THE 


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f  COLUMBIA]^  CALCULATOR; 


A  PRACTICAL  AND  COxNCISE  SYSTEM 


DECIMAL  ARITHMETIC. 


VHE  USE  OF  SCHOOLS  IN  THE  UNITED  STATES, 


BY    ALMOIN    TiO"  NOR, 


r  -'lUhA.   -ouNTANT's  .^^sts-^'ant;'  -math-^:^''- 


T  A  r;  r  K  :     '     7'  T  C 


"Such  beiug  ino  nature  of  Federal  money,  its  operations  can  in  no  way  be  so  ; 
well  understood  as  in  obtaining  a  good  knowledge  of  Decimals,  and  applying  ! 
Ll>eir  I'everul  rules  to  th\!  various  cases  of  money  matters." — Prof.  Dewey. 


Fourr^  '<   E[K  ;mo^-   ri;  ^'* 


/ND    CORRECTED. 


PHILADELPHIA: 
K  AY    &    TRO  UTM  A  N, 

I^ANIELS  &  SMITl  ,  AND  W.  A.  1>EARV  &  CO.  PITTSBURGH  :  ELLIOTT 
&  ExVGLlSH.  Ne\/  YORK:  J.  S.  REDFIELDj  HUNTING  JON^  'AVAGE,. 
AND  GATF/i  &  STEDMAN.  ALBANY:  E.  H.  PEASfe.  6^  ^0.  '^iUFFALO  .* 
O.  Kr.  STEELE.  CINCINNATI  :  C.  D.  TRC^MAN.  RICHMOND,  VA.".  J. 
W.  RANDOLPH  ^  CO,       ;  ''■  :  HALL  &  DICKSON. 


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THE 


COLUMBIAI  CALCULATOR 


BEING 


A  PRACTICAL  AND  CONCISE  STSTEIM 


f 


DECIMAL  ARITHMETIC. 


APAPTED  TO 


THE  USE  OF  SCHOOLS  IN  THE  UNITED  STATES. 


BY    ALMON    TICKNOR, 

AUTHOR  OF   "THE  accountant's  ASSISTANT,"   "  MATHEMATICAL  TABLES,"   ETC. 


"  Such  being  the  nature  of  Federal  money,  its  operations  can  in  no  way  be  so 
well  understood  as  in  obtaining  a  good  knowledge  of  Decimals,  and  applying 
their  several  rules  to  the  various  cases  of  money  matters." — Prof.  Dewey. 


FOURTH    EDITION.  REVISED    AND    CORHECTED. 


PHILADELPHIA; 
KAY   &   T  R  O  U  T  M  A  N , 

DANIELS&SMITH,  ANDW.A.LEARY  AND  CO.  PITTSBURGH:  ELLIOTT 
<fc  KNGLISH.  NEW  YORK!  J.  S.  REDFIELD,  HUNTINCDON  i^J  SAVAGE. 
AND  GATES  &  STEDMAN.  ALBANY;  E.  H.  PEASE  &  CO.  BUFFALO: 
O.G.STEELE.  CINCINNATI:  E.  D.  TRUMAN.  RICHMOND,  VA. :  J. 
W.  RANDOLPH  &  CO.       SYRACUSE:    HALL  &  DICKSON. 


RECOMMENDATIONS. 

From  W.  McCartney,  Esq,,  Professor  of  Mathematics,  Laf ay » 
ette  College. 

,,     rp,  E ASTON,  January  7,  1845. 

Mr.  Ticknor  : — 

Dear  Sir :  I  have  looked  over  some  of  the  proof-sheets  of 
your  treatise  on  Arithmetic,  and  am  pleased  to  observe  that  you 
have  introduced  many  practical  examples  in  illustration  of  the 
rules.  Your  book  is  well  adapted  to  those  who  desire  a  prac- 
tical work  on  the  subject,  and  is  full  in  details  and  illustrations 
for  those  who  are  commencing  the  study  of  this  science.  Prac- 
tical books  are  the  kind  adapted  to  the  business  transactions  of 
the  acre.  Very  truly,  yours,  &c. 

w.  McCartney. 


From  D.  P.  Yeomans,  Professor  of  Chemistry  and  Principal  of 
the  Model  School  in  Lafayette  College. 

Easton,  February  1,  1845. 
I  have  examined  the  work  entitled  the  "  Columbian  Calcu- 
lator," by  Mr.  Ticknor,  and  deem  it  well  adapted  as  a  work  on 
practical  Arithmetic  for  use  in  common  schools.  The  numerous 
examples  employed  to  illustrate  principles,  will  render  it,  in  the 
hands  of  competent  instructors,  peculiarly  valuable,  to  both  the 
student  and  the  man  of  business.  D.  P.  YEOMANS. 


From  N.  Olmstead,  Teacher  of  a  Public  School  in  Easton,  Pa, 

Mr.  Ticknor  :-  ^*^^°^'  February,  1845. 

Dear  Sir  :  From  a  pretty  thorough  examination  of  your  sys^ 
tem  of  Arithmetic,  I  can  say  without  hesitation,  that  in  my 
opinion  it  is  decidedly  superior,  for  the  use  of  common  schools, 
la  any  now  in  use.  The  currency  of  our  country,  in  every  sys 
tem  of  Arithmetic,  should  be  of  paramount  importance ;  and  ir 
this  respect,  I  think  your  system  may  challenge  competition. 
Yours,  &c.  NICHOLAS  OLMSTEAD. 


Entered,  according  to  Act  of  Congress,  in  the  year  1846.  by  Almon  Ticknor, 
m  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Eastern 
District  of  Pennsylvania. 

STEREOTYPED  BY   REDFIELti   &   SAVAGE,   NEW  YORK. 


REMARKS. 

As  the  first  edition  of  this  work  was  favorably  received  by 
the  public,  notwithstanding  its  deficiency  and  imperfections,  it 
is  considered  a  sufiicient  inducement  to  enlarge  and  improve 
the  present  volume,  and  present  it  to  those  for  whom  it  is  in- 
tended, in  a  more  perfect  and  acceptable  form,  by  the  introduc- 
tion of  a  variety  of  rules,  examples,  questions,  and  reviews, 
arranged  and  explained  in  a  manner  that  the  author  feels  assured 
will  meet  the  general  approbation  of  the  competent  and  unprej- 
udiced teacher.  The  approbation  of  a  majority  of  popular 
teachers  and  other  persons  of  the  first  respectability,  in  relation 
to  the  arrangement  and  selection  of  suitable  and  instructive 
questions,  is  satisfactory  evidence  that  the  utility  and  merits  of 
the  work  have  been  duly  appreciated,  when  it  is  considered  that 
the  country  is  already  abundantly  supplied  with  Arithmetics, 
which  have  been  adopted,  and  claim  precedence  and  superiority 
over  other  treatises  of  this  description.  The  author  is  now 
more  fully  confirmed  in  his  belief  than  ever,  that  this  system  is 
the  only  true  and  correct  method  of  communicating  direct  and 
correct  instruction  in  the  science  of  numbers  ;  and  the  time  is 
not  far  distant  when  this  opinion  will  be  unanimous  throughout 
the  country,  that  a  systematic  arrangement  of  the  rules,  with  a 
thorough  knowledge,  both  theoretically  and  practically,  of  their 
operations,  together  with  a  variety  of  well-selected,  practical, 
rational  questions  for  solution,  is  of  the  first  importance,  and 
absolutely  necessary,  in  order  to  acquire  an  exact  and  expe- 
ditious method  of  calculation. 

As  this  edition  embraces  about  2,200  examples,  or  questions 
for  solution,  the  belief  is  entertained  that  this  number  will  be  as 
many  as  are  required  or  will  prove  beneficial  for  the  exercise 
of  the  pupil,  at  the  same  time  having  in  view  the  expense  of  the 
work,  which  it  is  desirable  should  be  placed  within  the  means 
of  every  one,  but  more  particularly  those  who  are  the  least  able 
to  procure  works  of  this  kind,  who  are  the  most  in  need,  and 
who  will  eventually  receive  the  most  benefit,  and  they  are  the 
persons  who  must,  and  will,  finally  preserve  and  perpetuate  the 
institutions  and  liberties  of  our  country.  The  greatest  care  and 
diligence  have  been  exerted  to  introduce  such  questions  as 
would  be  within  the  range  of  possibility,  and  likely  to  occur  in 
ihe  course  of  business,  and  in  adapting  the  language  and  phrase 


REMARKS. 


ology  to  the  occasion,  and  capable  of  being  correctly  understood 
by  those  for  whom  it  is  intended. 

Comparative  Arithmetic^  Analogy  ('*  resemblance  between 
things  with  regard  to  circumstances  or  effects,")  the  inductive 
'principle^  persuasive  or  mejital  Arithmetic — these  terms  are 
nearly  synonymous,  and  may  with  propriety  be  so  considered. 
The  idea  that  mathematics  can  be  taught  to  any  extent  by 
analogy f  induction,  or  degrees  of  comparison,  without  rules 
and  actual  calculation,  merely  by  the  effort  or  operation  of 
the  mind,  without  the  use  of  figures,  is  unquestionably 
absurd  and  inconsistent  in  the  extreme  ;  and,  strange  it  may 
appear,  some  authors  and  teachers  advocate  the  principle, 
and  call  it  precocity  or  march  of  intellect ;  the  system  is  neither 
founded  in  common  sense,  nor  on  philosophical  or  mathematical 
principles,  for  it  is  self-evident  that  every  effect  must  be  pro- 
duced by  some  corresponding  or  sufficient  cause  :  hence  arises 
the  necessity  of  rules  or  data,  on  which  to  found  a  system  of 
calculations,  which  must  operate  with  so  much  certainty  that 
they  shall  self-evidently  prove  their  own  theory  and  correctness, 
or  which  carry  their  own  evidence  with  them — the  eternal  prin- 
ciple of  truth ;  this  can  only  be  done  by  actual  calculation — by 
induction  never.  Can  the  surveyor  calculate  the  different 
angles  of  his  survey  by  analogy,  and  without  the  use  of  decimal 
tables,  and  the  application  of  rules,  prepared  for  that  express 
purpose  ?  Can  the  navigator  take  solar  and  lunar  observations, 
and  calculate  the  places,  southing,  and  setting  of  the  stars,  on  the 
inductive  principle  ?  Can  the  astronomer  from  analogy  (induc- 
tion), and  without  figures,  trace  the  abstruse  paths  or  parabolic 
orbits  of  the  comets,  and  calculate  the  powers  of  attraction  and 
repulsion,  the  distances  and  motions  of  the  planetary  system  ? 
Does  not  every  mathematician  know,  that  the  sciences  mentioned 
above  require  the  most  intense  study,  and  the  application  of 
certain  unerring  rules  ? 

A  variety  of  questions  and  examples  have  been  given  in 
square  and  cubical  measure,  mensuration,  etc.,  which  will  be 
found  highly  useful  and  entertaining  to  the  pupil,  and  in  num- 
bers sufficient  for  the  usual  occurrences  of  practical  business  life. 

A  free  use  of  the  black-board  in  school  is  earnestly  recom- 
mended, to  make  the  ready  calculator. 
j!      To  those  who  have  used  or  recommended   the  preceding 
(  edition  of  this  work,  the  author  would  embrace  the  present  op- 
portunity of  tendering  his   sincere  acknowledgments,  for  the 
kind  and  obliging  manner  uniformly  manifested  in  his  behalf. 

A  Key  to  this  work  is  now  published. 


A  SHORT  INTRODUCTION  TO  AMERICAN  CURRENCY 

When  this  country  was  subject  to  the  crown  of  England,  we 
were  governed  by  their  laws,  and  adopted  their  method  of  reck- 
oning money,  in  pounds,  shillings,  and  pence,  which  differed  in 
value  in  the  several  colonies,  as  they  were  then  called,  but  Lre 
now  called  states  ;  neither  was  it  of  the  same  value  of  the 
currency  of  England.  The  pound  currency  of  England  was 
once  equal  to  a  pound  avoirdupois,  which  was  many  times  its 
present  value.  Under  the  colonial  government,  the  several 
states  issued  bills  of  credit  to  supply  the  want  of  specie,  and 
to  answer  a  medium  of  trade ;  but  as  these  bills  were  not  re- 
ceived by  the  British  merchants  in  payment  for  goods  at  their 
par  value,  holders  of  the  bills  had  to  pay  more.  Thus,  in  New 
York  they  had  to  pay  in  the  bills  of  the  state  at  the  rate  of  8 
shillings  for  4  shillings  and  6  pence  sterling,  and  so  in  propor- 
tion to  the  depreciation  of  the  bills  of  the  other  states.  Taking 
4  shillings  6  pence  sterling  as  the  value  of  a  dollar,  the  currency 
of  the  New  England  states  was  6  shillings  to  the  dollar :  or,  6 
shillings  New  England  currency  was  worth  4  shillings  and  6 
pence  English  currency ;  New  York  and  North  Carolina,  8 
shillings  to  a  dollar,  or  equal  to  4  shillings  and  6  pence  Eng- 
lish ;  Virginia,  Kentucky,  Tennessee,  and  Ohio,  6  shillings ; 
New  Jersey,  Pennsylvania,  Delaware,  and  Maryland,  7  shillings 
and  6  pence ;  South  Carolina  and  Georgia,  4  shillings  and  8 
pence  ;  (Canada  and  Nova  Scotia,  5  shillings).  This  state  of 
the  currency  was  attended  with  much  inconvenience,  and  is  at 
the  present  day,  but  is  fast  going  into  disuse,  much  to  the  relief 
of  the  community.  After  the  war  of  the  revolution  we  became  a 
separate  and  independent  nation,  framed  our  own  laws,  coined 
money,  fixed  the  value  of  currency,  &c.  Our  coins  are  of  gold, 
silver,  and  copper,  and  their  weight  and  value  so  proportioned 
as  to  increase  from  the  lowest  to  the  highest  by  tens,  or  in  a 
tenfold  proportion ;  that  is,  ten  of  every  lower  denomination,  or 
less  value,  make  one  of  the  next  higher,  and  consequently  one 
of  every  higher  makes  ten  of  the  next  lower.  Thus  we  say, 
10  mills  make,  or  are  equal  to  1  cent  in  value;  10  cents  are 
equal  to  1  dime;  10  dimes  are  equal  to  1  dollar;  and  10  dol- 
lars are,  equal  to  1  eagle.  There  are  also  the  half  cent,  half 
dime,  quarter  and  half  dollar,  quarter  and  half  eagle ;   but  in. 

1* 


0  A  SHORT   IXTIIODUCTION   TO   AMHRICAX  CUI^REXCY. 

reckoning  money,  ir  is  customary  to  use  dollars,  cents,  and 
mills,  omitting  the  other  denominations.  In  writing  numbers  in 
dollars,  cents,  and  mills,  we  leave  a  small  space  between  them, 
or  separate  them  with  a  comma  (,)  : — 

V'  25  '"5''"  five  dollars  twenty-five  cents  and  five  mills  ; 
dollars  5,  25,  5  ;  or  dollars  5,  2,  5,  5 — five  dollars  two  dimes 
five  cents  and  five  mills. 

The  rule  for  adding  several  sums  together  in  dollars,  cents 
and  mills,  is  this  : — add  all  the  mills,  and  carry  one  for  every 
ten,  and  add  this  one  to  the  next  figure  in  the  place  of  cents ; 
but  if  it  is  less  than  ten,  set  it  down  directly  under  the  column 
of  mills,  then  add  the  cents  in  the  first  column,  and  carry  one 
for  every  ten  to  the  next  column  of  cents,  which  add,  and  carry 
one  for  every  ten,  as  before,  to  the  place  of  dollars  ;  add  the  dol- 
lars and  set  down  the  full  amount ;  for  even  tens  write  a  cipher 
(0).  If  A  owes  you  dollars  5,  25,  5,  and  B  dollars  8,  82,  9, 
how  much  do  both  owe  you?     A     5,  25,  5 

B     8,  82,  9 

14,  08,  4 

In  this  example,  we  say  9  and  5  are  14,  set  down  4,  and 
carry  1  to  the  next  figure  2,  which  will  make  3,  and  5  are  8 ; 
set  down  the  8,  and  nothing  to  carry,  because  it  is  less  than  10  ; 
then  8  and  2  are  10,  set  down  a  cipher,  because  it  is  even  (10), 
and  carry  1  to  the  next  figure  8,  will  make  9,  and  5  are  14 ;  set 
down  all  the  last  column,  so  that  the  two  sums  owing  by  A  and 
B,  when  added  together,  will  make  dollars  14,  08,  4 — fourteen 
dollars,  eight  cents,  and  four  mills. 

Note. — No  scholar  should  be  permitted  the  use  of  an  arith- 
metic until  he  has  been  exercised  in  the  first  four  rules,  both 
mentally  and  on  the  hoard ;  every  school  should  have  several 
boards,  one  for  each  class,  and  not  let  a  day  pass  without  their 
use,  by  having  the  teacher  give  examples  first,  then  the  pupils, 
beginning  with  the  first,  and  continue  through  the  class,  without 
the  book ;  prove  their  questions,  and  give  their  reasons :  this 
will  exercise  the  mind,  and  bring  forward  all  the  faculties ;  in 
this  way,  there  will  be  more  knowledge  acquired  in  one  month 
than  there  can  be  in  ten  in  conning  over  a  book,  which  is  a 
blank  to  them,  and  but  little  better  to  their  teachers.  It  M^ill 
attract  attention  and  excite  emulation,  more  than  any  other 
course,  which  is  the  mainspring  in  the  acquisition  of  knowledge 

' — STRICT    ATTENTION. 


TABLE    OF    CONTENTS. 


V  PAOB. 

Simple  Numbers,  from         -        -        -        -        -        -9  to  48 

Decimal  Fractions 43  to  59 

Tables  of  Weights  and  Measures         -----  60 

Simple  Reduction 66 

Practical  Questions 70 

Compound  Numbers 72  to  87 

Reduction  of  Decimals         --*-.--  88 

Proportion    ----------  101 

Vulgar  Fractions 114  to  135 

Practice 136 

Simple  Interest --  140 

Coins,  Currency,  &c.    -        -        - 159 

Discount      ----------  162 

Equation 166 

Barter 168 

Profit  and  Loss 172 

Partnership 177 

Taxing 179 

Percentage,  do.  on  Lands     -        -        -        -        -        -        -183 

Gross,  Tare,  &c. --.-  185 

United  States  Duties 187 

Position 188 

Involution    --*----•--  194 

Evolution 195 

Square  Root .--  195 

Cube  Root 202 

Allegation 208 

Progression  -------                         -  212 

Permutation 219 

Combination         ---------  220 

Compound  Interest       --------  221 

Annuities -----  225 

Duodecimals         ---- 23C 

Appendix     ---,------  235 

Artificers'  Work 235  to  240 

Mensuration ----  240 

Miscellaneous  Matter  in  Mensuration  -----  249 

Promiscuous  Questions          -------  258 

Black-board 363 


EXPLANATION  OF  CHARACTERS. 

Signs.  Significations. 

=  Equal ;  as,  10  mills  =  1  cent,  10  cents  =  1  dimCj  10  dimes  =  1 
dollar,  10  dollars  =  1  eagle;  and  when  placed  between  two 
numbers  it  denotes  that  they  are  equal  to  each  other. 

+  More,  or  addition;  as,  6+2  =  8,  3-f-2  =  5;  when  the  num- 
bers are  small,  we  can  say  6  and  2  are  8,  3  and  2  are  5. 
This  sign  is  sometimes  placed  at  the  right  of  the  quotient,  or 
answer,  denoting  that  there  is  a  small  remainder.  It  is  also 
called  plus,  meaning  more. 

—  Less  (minus),  or  subtraction ;  when  placed  between  two  num- 
bers, it  denotes  that  the  one  on  the  right  is  to  be  taken  from 
the  one  on  the  left;  thus,  5  —  3  =  2  denotes  that  3  is  to  be 
taken  from  5,  will  =2,  or  2  over,  which  is  the  remainder ;  5 
is  the  minuend,  and  3  the  subtrahend  ;  when  the  numbers  are 
small,  you  can  say  3  from  5  leaves  2.  It  is  sometimes  used 
in  division,  signifying  a  remainder,  as  2  in  5,  2  —  1  remain- 
der ;  2  in  5  twice  and  1  over. 

X  Into,  or  multiplied  by,  sign  of  multiplication  ;  thus,  4x4=16, 

5  X  5=125,  denotes  when  placed  between  two  numbers  that 
they  are  to  be  multiplied  together ;  6  X  6  =  36,  that  is,  6  times 

6  are  36.     X  is  sometimes  used. 

•r  Division,  or  divided  by;  as, 25-^-5  =  5,  or  8)64(8,  or  ^8^  =  8, 
meaning  that  25  divided  by  5,  the  answer  will  be  5,  :  64 
divided  by  8  =  8  answer  :  again,  ^-^  denotes  that  64  is  divided, 
8  :  and  8)^3^  signifies  that  64  is  divided  by  8.  E.  eagle,  D. 
or  $,  dollar,  d.  dime,  c.  or  cts.  cents,  m.  mills. 

f  Period,  or  decimal  point,  separatrix,  used  to  distinguish  deci- 
mals from  whole  numbers,  as  those  on  the  right  are  decimals, 
and  on  the  left  (if  any)  are  whole  numbers ;  thus,  .25  ;  46 
.24  :  it  is  used  to  separate  D.  c.  m. ;  the  comma  (,)  is  also 
used  for  this  purpose. 

:  ::  :  Proportion ;  as,  2  :  4  ::  8  :  16,  that  is,  as  2  is  to  4,  so  is  8 
to  16  ;  or  the  same  proportion  that  4  is  to  2,  so  is  16  to  8. 

12 — 3  +  5=4,  a  vinculum;  the  line  over  the  3  and  5  connects, 
all  the  numbers  over  which  it  is  drawn  as  simple  numbers, 
as  3  and  5  are  8,  from  12=4. 

-/  or  2/  Square  Root;  as,  ^64  =  8:  denotes  that  the  square 
root  of  64  is  8. 

y  Cube  Root;  as,  ^64=4  :  ^  Biquadrate  Root,  as  ^64=2. 
Dollars,  dimes,  and  mills  increase,  and  are  calculated  the 

same  as  whole  numbers. 


THE  COLUMBIAN  CALCULATOR. 


ARITHMETIC. 

Arithmetic  is  a  part  of  the  science  of  Mathematics,  and  ia 
the  art  of  computing  by  numbers,  by  the  operation  of  six  rules, 
namely,  Notation,  Numeration,  Addition,  Subtraction,  Multipli- 
cation, and  Division ;  two  of  which  may  be  considered  primary 
rules,  namely,  Addition  and  Subtraction,  and  the  other  four  sec- 
ondary, as  they  naturally  arise  from  the  operation  of  the  former. 
The  knowledge  of  this  science  is  so  universally  necessary,  that 
scarcely  anything  in  life,  and  nothing  in  trade,  can  bo  done 
without  it. 


NOTATION. 

Notation  teaches  to  express  words,  or  numbers,  by  ten  Arabic 
characters,  or  digits,  namely,  1,  one  ;  2,  two  ;  3,  three  ;  4,  four ; 
5,  five  ;  6,  six  ;  7,  seven  ;  8,  eight ;  9,  nine  ;  0,  cipher  ;  by  the 
use  of  which,  all  numbers  are  expressed,  and  increase  in  value 
from  right  to  left,  in  a  tenfold  proportion  ;  thus,  1824,  the  figure 
(4)  in  the  place  of  units,  denotes  only  its  simple  value,  4  ;  that 
in  the  second  place,  or  place  of  tens  (2),  is  ten  times  its  simple 
value,  24  ;  that  in  the  third  place,  or  place  of  hundreds  (8),  one 
hundred  times  its  simple  value,  824  ;  that  in  the  fourth  place,  or 
place  of  thousands  (1),  one  thousand  times  its  simple  value,  or 
1824,  one  thousand,  eight  hundred  and  twenty-four. 

A  cipher  when  alone  is  of  no  value,  but  when  placed  to  the 
right  of  a  figure,  increases  the  value  of  that  figure  in  a  tenfold 
proportion ;  thus,  5  alone  is  only  five ;  but  annex  a  cipher  to  the 
tight,  thus,  50,  and  it  increases  the  value  tenfold,  or  fifty. 


10 


NOTATION. 


EXAMPLES. 

Write  the  following  numbers  in  figures  :  twenty-five  ;  seven* 
ty-six  ;  ninety-one  ;  eighty-four  ;  sixty-five  ,  nineteen  ;  eleven  ; 
one  hundred ;    one    hundred   and  sixty-seven ;    two   hundred 
four  hundred  ;  one  thousand  ;  two  thousand,  nine  hundred  and 
ninety-nine. 

In  the  Roman  method  of  notation  by  letters,  I  represents 
one ;  V,  five  ;  X,  ten  ;  L,  fifty  ;  C,  one  hundred  ;  D,  five  hun- 
dred ;  M,  one  thousand,  &c. 

As  often  as  any  letter  is  repeated,  so  many  times  its  value  is 
repeated,  unless  it  be  a  letter  representing  a  less  number,  placed 
before  one  representing  a  greater^  then  the  less  number  is  taken 
from  the  greater  ;  thus,  IV  represents  four,  IX,  nine,  &;c.,  as 
will  be  seen  in  the  following  table,  which  the  pupil  should  com- 
mit to  memory. 

ROMAN  METHOD,  OR  NOTATION  BY  LETTERS. 


One 

I                     Ninety 

XC 

Two 

II                   One  hundred 

c 

Three 

III                  Two  hundred 

cc 

Four 

IIII,  or  IV    Three  hundred 

COG 

Five 

V                   Four  hundred 

CCCC 

Six 

VI                  Five  hundred 

D,  or  io» 

Seven 

VII                Six  hundred 

DC 

Eight 

VIII               Seven  hundred 

DCC 

Nine 

Villi, or  IX  Eight  hundred 

DCCC 

Ten 

X                   Nine  hundred 

DCCCC 

Twenty 

XX               One  thousand 

M,  or  Clot 

Thirty 

XXX            Five  thousand 

100.  or  ^X 

Forty 

XL               Ten  thousand 

CCIqo,  or  X 

Fifty 

L                  Fifty  thousand 

1000 

Sixty 

LX               Hundred  thousand 

CCCIqoo,  or  C 

Seventy 

LXX            One  million 

M 

Eighty 

LXXX         Two  millions 

mm: 

MDCCCXLIV  =  A.  D.  1844 

•  l3  is  used  instead  of  D,  to  represent  five  hundred,  and  for  every  ad- 
ditional (3  annexed  at  the  right,  the  number  is  increased  ten  times. 

t  CI  J  is  used  to  represent  one  thousand,  and  for  every  C  and  q  put  at 
each  end,  the  number  is  increased  ten  times. 

X  A  line  over  any  number  increases  Its  value  one  thousand  times. 


NUMERATION. 


11 


NUMERATION. 

By  Numeration  we  are  taught  to  read  any  number  of  figures, 
and  ascertain  their  relative  value,  when  taken  in  connexion  with 
each  other,  which  is  determined  by  the  situation  in  which  they 
are  placed,  and  more  correctly  and  perfectly  exemplified  by  the 
following  tables : — 


# 

1 

TABLE 

I. 

CO 

S 

o 

CO 

0 

CQ 

^ 

. 

o 

'^ 

*a 

CO 

g 

rg 

q 

rt 

CQ 

. 

o 

CO 

t3 

CO 

r^ 

CO 

T3 

p 

fl 

cd 

t3 

<o 

r3 

^ 

? 

o 

2 

•5 

s 

o 

V4 

-5 

1 

M 

fl 

s 

3 

1 

fl 

3 

i 

rC3 

9 

8 

7 

6 

5 

TABLE 

4 
11 

3 

1  one. 
12  twelve, 
123  1  hundred  23. 
1234  1  thousand  2  hundred  and  34. 
12345  12  thousand  3  hundred  and  45. 
123456  123  thousand  4  hundred  and  56, 
1234567  1  million  234  thousand  5  hundred  and  67. 
12345678  12  millions  345  thousand  6  hundred  and  78. 
123456789  123  millions  456  thousand  7  hundred  and  89 


To  enumerate  figures  where  the  numbers  are  large,  it  will  be 
found  convenient  to  divide,  or  separate  them,  into  periods  of 
three  figures,  the  first  being  hundreds,  the  second,  hundreds  of 
thousands,  &c.  Then  begin  at  the  right  hand,  or  place  of  units, 
and  read  toward  the  left,  as  in  table  1st;  thus  1  is  in  the  place 
of  units,  2  is  in  the  place  of  tens,  and  3  in  the  place  ef  hun- 
dreds, which  three  figures,  taken  in  connexion,  would  express 
(321)  three  hundred  and  twenty  one. 

In  table  2d,  the  numbers  may  be  enumerated  in  the  same 
way,  and  then  read  from  left  to  right,  in  the  order  in  which  they 
stand. 


12  NUMERATION. 

TABLE    III. 

1  units, 
12  tens. 
123  hundreds. 
1234  thousands. 
12345  tens  of  thousands. 
123456  hundreds  ;/  thousands. 
1234567  millions. 
12345678  tens  of  millions. 
123456789  hundreds  of  millions. 
1234567891  thousands  of  millions. 
12345678912  tens  of  thousands  of  millions. 
1*23456789123  hundreds  of  thousands  of  millions. 
1234567891234  billions. 
12345678912345  tens  of  billions. 
123456789123456  hundreds  of  billions. 
1234567891234567  thousands  of  billions. 
12345678912345678  tens  of  thousands  of  billions. 
}  23456789123456789  hundreds  of  thousands  of  billions. 

Quintillions.         Quadrillions.         Trillions. 
555,555  555,555  555,555 

Nonillions.         Octillions.         Septillions.         Sectillions. 
555,555  555,555  555,555  555,555 

The  third  table  is  given  to  express  the  higher  powers,  and 
«an  be  read  and  applied  in  the  same  manner  as  the  two  prece- 
ding tables.  When  even  hundreds,  thousands,  &c.,  are  to  be 
written,  the  vacant  places  of  units,  tens,  hundreds,  &;c.,  are  to 
be  supplied  by  ciphers  ;  thus,  to  write  one  thousand  (1000), 
place  three  ciphers  to  the  right  of  the  1,  and  so  in  all  cases  of 
a  similar  nature. 

EXAMPLieg. 

Write  down  in  words,  the  following  numbers  :  10,  15,  20,  25, 
30,  35,  40,  45,  50,  55,  60,  65,  70,  75,  80,  85,  90,  95,  100,  150, 
476,  891,  999,  1001.  Let  the  following  numbers  be  expressed 
in  figures :  twenty,  thirty,  forty-five,  seventy-six,  eighty-three, 
ninety-one,  one  hundred  and  one,  one  hundred  and  nine,  one 
hundred  and  fifteen,  one  hundred  and  thirty,  one  hundred  and 
ninety-four,  two  hundred  and  sixty,  three  hundred  and  forty-five, 
five  hundred  and  eighty-five,  six  hundred  and  ninety-one,  on© 
hundred  and  fifty  thousand. 


NUMERATION.  13 

The  preceding  is  the  English  method  of  Numeration ;  the 
following  is  the  French.  The  teacher  can  use  either  at  his 
pleasure : 

r " > 

Quadrillions.  Trillions.  Billions.   Millions.  Thousands.  Units. 
321  321  321  321  321  321 


What  is  Arithmetic?  By  what  means  are  operations  i 
Arithmetic  performed  ? — Name  "them.  What  is  Notation  : 
What  is  Numeration  ?  How  must  figures  be  enumerated  ?  In 
what  manner  should  figures  be  read  ?  Why  do  you  enumerate 
from  right  to  left?  In  what  proportion  do  they  increase  in 
value  ?  Recite  the  Numeration  Table.  Write  down  seventeen 
millions  :  —  seventeen  hundred  thousand. — Eleven  billions 
and  nine  hundred  thousand. 


SIMPLE    ADDITION. 

Addition  is  the  first  primary  rule  in  Arithmetic,  the  use  of 
which  is  to  ascertain  the  amount  or  sum  total  of  two  or  more 
numbers,  when  put  or  added  together  ;  as,  5 +  5  ==10:  that  is 
5  and  5  make  10. 

RULE. 

Set  the  given  numbers  under  each  other,  with  units  under 
units,  tens  under  tens,  hundreds  under  hundreds,  &:c.  Then 
draw  a  line  under  the  lowest  number,  and  begin  at  the  right 
hand  column,  or  place  of  units,  and  add  (upward)  all  the  column 
together ;  set  down  the  sum  when  less  than  ten  ;  if  ten  or  more, 
set  down  the  right  hand  figure,  and  add  the  left  hand  figure  to 
the  next  column ;  and  thus  proceed  to  the  last  column,  and  set 
down  the  wholes  amount  of  it. 

PROOF. 

Perform  the  operation  a  second  time,  agreeably  to  the  Rule ; 
but  in  this  case,  begin  at  the  top ;  or  reserve  one  of  the  given 
numbers,  find  the  sum  of  the  rest,  and  thereto  add  the  number 
reserved. 

Note. — The  reason  why  you  carry  one  for  every  ten,  is  this  • 
in  the  place  of  units,  it  requires  ten  to  make  one  in  the  place 
of  tens ;  and  in  the  place  of  tens,  it  requires  ten  to  make  one 
in  the  place  of  hundreds  ;  therefore,  you  always  carry  one  from 
one  denomination  to  another,  as  it  requires  of  that  denomination 
to  make  one  in  the  next. 

2 


SIMPLE    ADDITION. 


ADDITION  TABLE. 


2  and  2 

are  4 

3  and  9 

are  12 

5  and  9  are  14 

8  and  8  are  16 

2    3 

5 

3 

10 

13 

5 

10 

15 

8 

9 

17 

2    4 

6 

3 

11 

14 

5 

11 

16 

8 

10 

18 

2    5 

7 

3 

12 

15 

5 

12 

17 

8 

11 

19 

2    6 

8 

4 

4 

8 

6 

6 

12 

8 

12 

20 

2    7 

9 

4 

5 

9 

6 

7 

13 

9 

9 

18 

2    8 

10 

4 

6 

10 

6 

8 

14 

9 

10 

19 

2    9 

11 

4 

7 

11 

6 

9 

15 

9 

11 

20 

2   10 

12 

4 

8 

12 

6 

10 

16 

9 

12 

21 

2   11 

13 

4 

9 

13 

6 

11 

17 

10 

10 

20 

2   12 

14 

4 

10 

14 

6 

32 

18 

10 

11 

21 

3    3 

6 

4 

11 

15 

7 

7 

14 

10 

12 

22 

3    4 

7 

4 

12 

16 

7 

8 

15 

11 

11 

22 

3    5 

8 

5 

5 

10 

7 

9 

16 

n 

12 

23 

3    6 

9 

5 

6 

11 

7 

10 

17 

12 

12 

24 

3    7 

10 

5 

7 

12 

7 

11 

18 

13 

13 

26 

3    8 

11 

5 

8 

13 

7 

12 

19 

14 

14 

28 

To  read  t 

he  t£ 

ible, 

say  2  . 

md  2 

are  4 

3  and  3 

are  6, 

&c. 

All  of  the  tables,  and  rules  generally,  should  be  committed  to 
memory,  before  the  pupil  attempts  the  solution  of  the  questions. 

QUESTIONS. 

2463  Explanation. — Begin  by  saying,  1  and  3  are 

4,  and  2  are  6,  and  3  are  9  ;  set  it  down.     Then, 

4532  4  and  7  are  11,  and  3  are  14,  and  6  are  20  ;  set 

0773  down  0,  and  carry  2  to  8,  which  will  make  10, 

9841  and  7  are  17,  and  5  are  22,  and  4  are  26  ;  set 

down  the  6,  and  carry  2  to  the  next  figure,  9, 

17609  amount,  which  will  make  11,  and  0  is  nothing,  but  4  are 

15,  and  2  are  17;   always  set  down  the  whole 

15146  amount  of  the  last  column.     Then,  to  obtain  the 

proof,  draw  a  line  under  the  numbers  at  the  top, 

17609  proof.      and  add  the  remaining  numbers  as  before,  omit- 
ting the  numbers  at  the  top,  and  you  find  this 

sum  to  be  15146  ;  this  last  sum  is  now  added  to  the  numbers 
at  the  top,  which  agrees  with  the  first  addition  ;  hence  it  is  sup- 
posed to  be  correct.  Note. — A  question  may  be  proven  accord- 
ing to  rule,  and  be  incorrect,  because  the  same  error  may  be 
committed  in  the  proof  that  was  in  the  first  calculation. 

1.  Henry  has  7  apples,  and  George  will  give  liim  11 ;  how 
many  will  he  then  have  ? 


SIMPLE    ADDITION. 


15 


2.  Thomas  has  17  cents,  John  19,  William  14;  how  many 
in  all? 

3.  Samuel  has  35  cents,  he  had  lost  6  ;  how  many  had  he  at 
first  ?     . 

4.  How  many  are  17,  18,  19,  20,  and  21  ?  How  many  are 
52,76,  83,  82,  2,  1,  and  17? 

5.  William  paid  for  a  knife  31  cents,  pocket-book  25  cents, 
%late  18  cents,  book  42  cents,  paper  7  cents  ;  what  did  they 
ail  cost  ? 

6.  A  farmer  has  19  cows,  157  sheep,  and  84  calves;  how 
rjany  in  all  ? 


8 


9 


10 


11 


12 


to 

J3 

3   § 

CO  w 

hun'ds, 

tens. 

units. 

Ij 

«3  n 
II 

4 

6 

3  7 

6  5 

8  6  4 

6  4 

3  2 

3 

3 

8  4 

8  7 

3  2  0 

1  5 

8  6 

5 

2 

6  5 

4  3 

1  7  6 

5  4 

9  6 

6 

4 

8  4 

2  9 

5  8  3 

7  8 

6  7 

7 

6 

0  7 

8  5 

4  7  6 

4  5 

7  6 

4 

8 

6  5 

7  6 

5  0  3 

6  8 

9  6 

0 

9 

8  0 

4  3 

9  6  8 

4  2 

5  3 

2 

7 

6  1 

2  1 

4  7  9 

2  1 

3  8 

1 

8 

4  8 

0  6 

6  4  0 

6  5 

6  4 

7 

6 

6  5 

5  4 

2  0  4 

2  1 

0  3 

13 

14 

15 

16 

17 

5314 

4673 

46785 

24605 

71456 

2302 

2145 

43123 

03123 

04523 

1435 

3210 

65320 

64321 

13072 

2134 

4532 

13042 

45671 

34562 

1435 

3245 

2 

56231 

31230 

53103 

12620 

17805 

24501 

168950 

176716 

18 

19 

20 

21 

22 

23 

24 

25 

^is. 

dolls 

sec'ds.  minutes,  hours. 

days. 

months,  years. 

78 

80 

464 

431 

841 

302 

197 

6784 

64 

19 

876 

782 

674 

407 

642 

5378 

41 

16 

221 

641 

531 

681 

396 

9605 

82 

42 

642 

421 

486 

764 

421 

7432 

99 

78 

381 

764 

798 

842 

789 

1867 

71 

97 

496 

491 

642 

596 

462 

5978 

16 


SIMPLE    ADDITION. 


26.  What  is  the  amount  of  427,  632,  781,  1001  4765 
32014?  Answer,  39620 

27.  Add  672,  1021,  846,  27,  4762,  7820,  and  48  together. 

Ans.  15196 

28.  Add  the  following  numbers,  5876,  7890,  6874,  9658, 
1234.  Ans.  31532. 

29.  W.,  in  the  collection  of  money,  received  of  B.  45  dolls., 
of  C.  74  dolls.,  of  D.  96  dolls.,  of  E.  121  dolls. ;  how  much 
did  he  receive  in  all  ?  Ans.  336  dollars. 

30.  B.  has  four  fields  ;  in  the  first  he  has  85  sheep,  in  the 
second  97,  in  the  third  142,  in  the  fourth  234  ;  how  many  has 
he  in  all  ? 

31.  C.  purchased  at  a  store,  76  pounds  of 
coflfee,  27  pounds  of  sugar,  36  pounds  of 
cheese,  40  pounds  of  salt,  9  pounds  of  tea,  4 
pounds  of  raisins,  and  2  pounds  of  spice  ;  how 


Ans.  558. 


37 


many  pounds  in  all  ? 


32 
4763 
2184 
5763 
7298 
6042 
5769 
6586 


33 

68978 
47697 
58321 
79642 
63426 
48967 
48302 


34 

21345 

7896 

214 

30 

2 

67 

478 


35 
76321 
49687 
24768 
21324 
13214 
13214 
10204 


Ans.  194. 

36 
64057 
32165 
48732 
17694 
21879 
36582 
14768 


rJ»  "^  b  J»    •   Oi 

O  J'^  O  O  rJ^  1:5 
O  O  O  O  O  S 

6 
3 
6 

7 
2 

4 

2 


2  0 


38405  415333  30032  208732  235877   3  19  6  110 


D. 

42 
187 

96 
176 
340 
180 
652 

96 


38 

dimes. 
7 
6 
7 
3 
6 
7 
3 
7 


m. 

4 
3 

1 
2 
7 
2 
1 
4 


39 
D.     c.  m. 

1879  25  5 


470  21 
6741  57 
4202  87 

118  50 

642  09 
19  20 

142  78 


40 
D.     c. 

1921  47 
1482  78 
1720  50 
1471  97 
1682  31 
1476  91 
2978  77 
9642  31 


41 
Miles. 
46890 
97642 
76960 
43219 
84298 
64967 
36899 
47947 


42 
Seconds. 
896747 
342651 
986742 
489764 
651876 
981974 
532763 
876490 


43.  Required  the  amount  of  the  following  sums;  five  thousand 
seven  hundred  and  eighty ;  seventy-nine  thousand  four  hundred 
and  forty-two  ;  eight  hundred  and  eighty-nine  thousand ;  one  mil* 


SIMPLE    ADDITION. 


17 


aon  ;  five  hundred ;  twenty-one  ;  nine  ;  seven  hundred  ;  twenty- 
one  thousand;  fourteen ;  one  hundred  and  nine  ?     Ans,  1996575. 


44 

45 

46 

47 

48 

486543 

9874658 

8479643 

78964532 

4798764 

213456 

6750234 

5896745 

47809214 

2156845 

789407 

8456796 

8421354 

87642305 

2156845 

801508 

5834978 

6743214 

78967408 

9764058 

764314 

7640857 

5679434 

97814234 

7428405 

827641 

6978574 

5307642 

70974584 

6489675 

583670 

8328976 

2131478 

64897984 
32487023 
32487023 

4287345 
1472021 
1472021 

4466539 

53865073  ^ 

42659510 

49 

50 

51 

52 

53 

67809782 

31829176 

879678 

5789630 

79857690 

31918976 

15896502 

527659 

2139567 

87235278 

52031607 

321 

768659 

8976387 

76897652 

65897031 

78652 

318796 

6705896 

13783658 

21317215 

65897 

528781 

3125789 

21356785 

78975873 

551789 

709152 

2157682 

10239653 

54.  A  man  commenced  a  journey,  and  travelled  9  days  ;  the 
first  day  he  travelled  34  miles,  and  after  that  he  gained  1  mile 
every  day ;  how  many  miles  did  he  travel  in  the  9  days  ? 

Ans,  342. 

55.  A  drover  purchased  the  following  drove,  namely:  102 
oxen,  for  which  he  paid  5764  dolls.;  176  cows  for  3784  dolls.; 
50  calves  for  250  dolls.;  420  sheep  for  960  dolls.;  of  how 
many  did  his  drove  consist,  and  how  much  did  he  pay  ? 

Ans.  the  drove  748,  paid  10758  D. 

56.  A  miller  purchased  of  a  farmer  10  bags  of  wheat,  which 
were  to  be  paid  for  by  weight,  which  was  as  follows :  No.  1, 
165  pounds  ;  No.  2,  168  ;  No.  3,  171  ;  No.  4,  175  ;  No.  5, 178  ; 
No.  6, 180;  No.  7,  182;  No.  8,185;  No.  9,  189  ;  No.  10,  190; 
how  many  pounds  of  grain  did  he  buy?  Ans.  1783  pounds. 

57.  The  plantation  of  a  gentleman  will  produce  in  one  sea- 
eon,  1500  bushels  of  wheat,  worth  1875D. ;  1350  bushels  of 
rye,  worth  1283D. ;  2134  bushels  of  barley,  worth  1696D. , 
450  bushels  of  corn  worth  275D. ;  and  876  bushels  of  oats 
worth  421 D. ;  required  the  number  of  bushels  of  grain,  and  the 
value?  Ans.  6310  bushels,  value  5550D. 

58.  What  number  of  dollars  are  in  6  bags,  each  bag  contain 
\\g  37542D.  Ans.  2^5252. 


19  SIMPLE    ADDITION. 

99.  There  is  a  valuable  farm,  one  quarter  of  which  is  worth 
15674D. ;  how  much  is  the  whole  farm  worth  ?      Ans,  62696D. 


60  61  62  63 

D.     c.    m.         D.     d.     c.  m.  389678  D. 


52 

76 

2 

178 

7 

6 

3 

58321 

586 

15 

78 

3 

6 

9 

8 

7 

76521 

7635 

17 

19 

7 

189 

6 

5 

8 

0586 

63528 

378 

17 

6 

25 

7 

3 

9 

216 

875697 

1837 

65 

7 

6 

3 

2 

1 

790 

5563216 

1316 

00 

9 

268 

7 

9 

8 

210 

11610780 

64.  Add  4764608,  407,  76,  46876541,  76084763,  together.    > 

Ans.  127726395. 

65.  Add  48,  96,  423,  8765,  6420,  4876,  904078,  together. 

Ans.  924706. 

66.  Add  1,  7,  6,  941,  784,  54204,  56476096,  together. 

Ans.  56532039. 

67.  Add  746796,  48,  50701,  30,  40,  17,  17645,  together. 

Ans.  815277 

68.  Add  9,  11,  642,  4786,  3000104,  68588479,  879643978," 
together. 


REVIEW. 

Which  is  the  first  primary  rule  in  Arithmetic  ?  How  many 
primary  rules  are  there  in  Arithmetic  ?  What  are  they  called  ? 
How  many  secondary  rules  are  there  1  Name  them.  Why 
are  they  called  secondary  ?  What  is  Addition  1  What  is  the 
use  of  Addition?  What  is  the  rule  for  writing  the  numbers? 
After  the  numbers  are  all  written,  how  do  you  proceed  ?  When 
you  have  found  the  amount  of  all  the  figures  in  the  left  or  last 
column,  what  will  you  do  ?  How  do  you  prove  the  correctness 
of  the  work  ?  When  you  have  proved  your  sum  according  to 
rule,  do  you  know  it  to  be  correct  ?  Why  do  you  carry  1  foi 
every  10  ?     Recite  the  Addition  table. 

69.  Ten  partners  have  each  87452D.  20c.  in  trade ;  what  if 
3ie  amount  of  their  capital  ? 

70.  Add  4444444,  5555555,  6666666,  7777777,  888888a 
together. 


SIMPLE    SUBTRACTION.  19 

SIMPLE    SUBTRACTION. 

This  is  the  second  primary  rule  in  Arithmetic,  and  is  the 
very  reverse  of  Addition.  It  teaches  to  take  a  less  number 
from  a  greater  of  the  same  name,  and  to  show  the  difference,  or 
remainder,  as  10 — 5,  5  remain;  that  is,  5  subtracted  from  10, 
it  will  leave  5,  which  is  the  difference. 


1.  Write  down  the  greatest  number  first,  then  write  the  less 
number  directly  under  it,  observing  to  place  units  under  units  ; 
tens  under  tens,  (fee. ;  draw  a  line  underneath. 

2.  Begin  with  the  units,  or  right-hand  figure,  and  subtract 
that  figure  from  the  figure  over  it,  and  set  down  the  difference. 

3.  When  the  figure  in  the  lower  number  is  more  than  the  one 
above  it,  you  must  subtract  from  10,  and  the  difference  between 
that  figure  and  10  must  be  added  to  the  figure  in  the  upper  num- 
ber, then  set  down  that  figure. 

4.  When  you  subtract  from  10,  carry  1,  and  add  it  to  the  next 
left-hand  figure  ;  proceed  in  this  manner  with  all  the  figures, 
and  the  number  thus  obtained  will  be  the  difference  between 
the  two  given  numbers. 

RULE    IT. 

After  stating  the  sum  as  above  directed,  then  if  either  of  the 
lower  figures  be  greater  than  the  upper  one,  conceive  10  to  be 
added  to  the  upper  figure  ;  then  take  the  lower  from  it,  and  set 
down  the  remainder.  When  10  is  thus  added  to  the  upper 
figure,  there  must  be  1  added  to  the  next  lower  figure. 

PROOF. 

Add  the  remainder,  or  difference,  to  the  less  number,  and 
their  sum  will  be  equal  to  the  greater  number. 

[It  is  of  the  greatest  importance  that  the  pupil  should  be 
thoroughly  exercised  in  the  primary  rules  previously  to  entering 
on  others ;  a  few  examples  are  given  in  money  for  exercise. 
As  the  calculations  can  be  the  same  as  whole  numbers,  that  sub- 
ject being  decimal,  it  necessarily  requires  a  knowledge  of  those 
rules  to  a  correct  understanding  of  our  currency,  which  will 
follow  Long  Division.] 


30 


SIMPLE    SUBTRACTION. 


SUBTRACTION  TABLE. 


1- 

-1=0 

3- 

-3=0 

5- 

-5=0 

7- 

-7=0 

9- 

-9=0 

2 

1 

4 

1 

6 

1 

8 

1 

10 

1 

3 

2 

5 

2 

7 

2 

9 

2 

11 

2 

4 

3 

6 

3 

8 

3 

10 

3 

12 

3 

5 

4 

7 

4 

9 

4 

11 

4 

13 

4 

6 

5 

8 

5 

10 

5 

12 

5 

14 

5 

7 

6 

9 

6 

11 

6 

13 

6 

15 

6 

8 

7 

10 

7 

12 

7 

14 

7 

16 

7 

9 

8 

11 

8 

13 

8 

15 

8 

17 

8 

10 

9 

12 

9 

14 

9 

16 

9 

18 

9 

11 

10 

13 

10 

15 

10 

17 

10 

19 

10 

12 

11 

14 

1] 

16 

11 

18 

11 

20 

11 

13 

12 

15 

12 

17 

12 

19 

12 

21 

12 

14 

13 

16 

13 

18 

13 

20 

13 

22 

13 

15 

14 

17 

14 

19 

14 

21 

14 

23 

14 

16 

15 

18 

15 

20 

15 

22 

15 

24 

15 

17 

16 

19 

16 

21 

16 

23 

16 

25 

16 

18 

17 

20 

17 

22 

17 

24 

17 

26 

17 

19 

18 

21 

18 

23 

18 

25 

18 

27 

18 

20 

19 

22 

19 

24 

19 

26 

19 

28 

19 

2- 

-2=0 

4- 

-4=0 

6- 

-6=0 

8- 

-8=0 

10-10=0 

3 

1 

5 

1 

7 

1 

9 

1 

11 

1 

4 

2 

6 

2 

8 

2 

10 

2 

12 

2 

5 

3 

7 

3 

9 

3 

11 

3 

13 

3 

6 

4 

8 

4 

10 

4 

12 

4 

14 

4 

7 

5 

9 

5 

11 

5 

13 

5 

15 

5 

8 

6 

10 

6 

12 

6 

14 

6 

16 

6 

9 

7 

11 

7 

13 

7 

15 

7 

17 

7 

10 

8 

12 

8 

14 

8 

16 

8 

18 

8 

11 

9 

13 

9 

15 

9 

17 

9 

19 

9 

12 

10 

14 

10 

16 

10 

18 

10 

20 

10 

13 

11 

15 

11 

17 

11 

19 

11 

21 

11 

14 

12 

16 

12 

18 

12 

20 

12 

22 

12 

15 

13 

17 

13 

19 

13 

21 

13 

23 

13 

16 

14 

18 

14 

20 

14 

22 

14 

24 

14 

17 

15 

19 

15 

21 

15 

23 

15 

25 

15 

18 

16 

20 

16 

22 

16 

24 

16 

26 

16 

19 

17 

21 

17 

23 

17 

25 

17 

27 

17 

20 

18 

22 

18 

24 

18 

26 

18 

28 

18 

To  read  the  table,  say  1  from  10  and  9  remain;  4  from  7 
and  3  remain. 


SIMPLE    SUBTRACTIOxN. 


21 


QUE'STIONS. 

From  47084  Explanation. — Begin  and  say,  2  from  4 

Take  23192"!  and  2  will  remain,  which  set  down;  then, 

<  +     you  can  not  take  9  from  8,  because  9  is  more 

Rem.  23892  J  than  8;  therefore  you  must  say  9  from  10 

and  1   will  remain,  which  added  to  8  will 

Proof 47084  make  9,  which  set  down;    now,  you  have 

borrowed  1,  which  you  must  pay  by  adding 
it  to  the  next  figure  at  the  left,  which  will  make  2  ;  then,  you 
can  not  take  2  from  0,  but  you  must  say  2  from  10  and  8  will 
remain  ;  you  now  have  one  to  carry  to  3,  which  will  make  4  ; 
then  say  4  from  7  and  3  remain ;  now,  you  have  none  to  carry, 
because  you  did  not  borrow ;  say  2  from  4  and  2  will  remain : 
then  add  the  two  lower  numbers  together  for  proof. 

1 .  William  had  32  marbles,  in  playing  he  lost  8 ;  how  many 
had  he  then  ? 

2.  Joseph  had  47  cents,  he  paid  18  at  the  store  ;  how  many 
remain  ? 

3.  Out  of  48  eggs,  John  sold  29  ;  how  many  had  he  remain- 
ing? 

4.  If  a  man  should  lose  25  dollars  out  of  75,  how  many 
remain  ? 


From  24532 
Take    12321 


6 
145632 
034411 


Diff.      12211       111221 


7 
478965 
367843 

111122 


Proof  24532   145632    478965 


8 
8764321 
6543210 

2221111 

8764321 


9 

7865435 
7653204 

212231 

7865435 


10.  1567053-— 1234730  rem.  332323 


11.  1579085—1381964 

12.  1100000—   1000 

13.  1000000—  999999 

14.  1111111—  999999 

15.  1794321—  147808 

16.  1963004—1837999 

17.  2478432—2189760 


26 
8476535 
7563258 


27 
964235 
217943 


197121 

1099000 

1 

111112 
1646513 

125005 

288672 

28 
1047658 
146873 


18.  8796784- 

19.  9431476- 

20.  1058760- 

21.  2764583- 

22.  3540506- 

23.  4116181- 

24.  7168097- 

25.  9708769- 

29 
8643587 
1234509 


-7960583 
-8942316 

-  165843 

-  181419 

-  335060 

-  201418  < 

-  671497 

-  701304 

30 

1047653 

926782 


913277    746292 


900785    7409078 


120871 


3t 


SIMPLE    SUBTRACTION. 


31 

4965832 
1234679 


33 

18176085 
465937 


33 

9684597 
4107965 


34 

846587 
100701 


35 

7800650 
7799990 


36.  A  had  2720  bushels  of  wheat,  he  disposed  of  1987  bush- 
els ;  how  many  bushels  will  remain?  Ans.  733. 

37.  A  man  in  building  a  house  that  will  require  47621  bricks 
has  received  21234  ;  how  many  more  will  be  required  ? 

Ans.  26387. 

38.  Subtract  4  from  a  million,  and  give  the  proof. 

39.  How  much  is  seven  thousand  seven  hundred  and  fifty-one 
greater  than  two  thousand  six  hundred  and  seventy-eight  1 

40.  From  one  million,  take  ninety-nine  thousand. 


41 
4693698076 
3729489635 


42 
58602973028764 
40098806709875 


43 
329865782198 
300982098709 


44 
4965786943 
1476879678 


45 
9789804023784 
2098479234678 


46 
74802136458 
68497683597 


47 
98673982011 
89658098701 


48 
6310618763980 
1329019876543 


49 
982119860753289 
765432098765420 


50 

5896478967 
4678569748 


51 
211898784267 
147989476984 


52 
4801213245630 
1302469785487 


53 
539864298670 
432986702100 

106877596570 


54 
9865432701198 
3762987502290 

6102445198908 


55 

86543298765 
29176807983 

57366490783 


56 
4027058426 
2134169874 


57 
980476589794 
807547867982 


58 
48976480111 
37766554223 


SIMPLE    SUBTRACTION.  33 

59.  A  has  4760D.,  he  will  pay  1786D. ;  B  has  6781  D.,  he 
will  pay  3789D.  ;  C.  has  7420D.,  he  will  pay  .4949  ;  how  much 
did  they  all  have  ai  first,  and  how  much  did  they  all  pay  ? 

Ans,,  they  had  18961D.,  paid  10524D. 

60.  A  gentleman  has  3  farms,  the  first  contains  421  acres  ; 
second,  687A.  ;  third,  582 A.  ;  the  first  is  worth  18968D.  ;  2d, 
31486D. ;  3d,  25600D. ;  how  many  acres  has  he  in  all  ;  and 
how  much  are  the  three  farms  worth  ;  and  if  he  should  sell  742 
acres,  how  many  would  remain  ? 

Ans.  He  has  in  all  1690A.  ;  worth  76054D. ;  will  remain 
948  acres,  after  selling  742  acres. 

61.  What  is  the  diflference  between  87904032  and  23040978  ? 

Ans,  64863054. 

62.  If  you  give  468D.  45cts.  for  159  bushels  of  wheat,  and 
789D.  55cts.  for  321  sheep;  and  you  dispose  of  87  bushels  of 
wheat  for  220D.,  and  180  sheep  for  350D.  ;  how  many  bush- 
els of  wheat  will  you  have  remaining  ?  and  how  many  sheep  ? 
how  much  money  will  you  receive  for  your  wheat  and  sheep  1 

Ans,  72  bushels  wheat  5  141  sheep  ;  receive  570D, 

ADDITION    AND    SUBTRACTION. 

63.  A  guardian  paid  his  ward  at  one  time  721 D.,  at  another, 
984D.,  and  again,  840D. ;  how  much  would  be  left  out  of  the 
estate,  which  was  but  2545D.  ? 

64.  A  gentleman  held  a  bond  of  1984D.  ;  he  received  at  one 
time  520D.,  at  another  time  640D.  ;  how  much  is  due  ? 

Ans,  824D. 

65.  If  I  add  430,  621,  7840,  21,  76,  1,  97,  17,  490,  and  then 
subtract  that  amount  from  964082,  what  will  remain  ? 

Ans.  954489. 

66.  A  loaned  B.  at  one  time  17D.,  at  another  time  84D.,  and 
at  another  time  500D.  ;  and  B.  paid  427D.  ;  how  much  is  due  ? 

Ans.  174D. 

67.  A  merchant  purchased  674  pipes  of  wine  for  87640D.  ; 
he  sold  485  pipes  for  75481D.;  how  many  pipes  has  he  re- 
maining, and  how  much  did  they  cost  him  ? 

Ans.  189  pipes  ;  cost  12159D. 

REVIEW. 

Name  the  second  primary  rule  in  arithmetic  ?  How  do  you 
write  down  numbers  in  this  rule  ?  What  next  ?  When  the 
figure  below  is  more  than  the  one  directly  over  it,  how  do  you 
proceed  ?     What  will  you  do  when  you  borrow^  or  subtract  from 


-24  SIMPLE    MULTIPLICATION. 

ten  ?     How  do  you  prove  subtraction  ?     For  what  purpose  is 
subtraction  used  ?     Repeat  the  rule  ? 

68.  Three  young  men,  A,  B,  and  C,  each  inherited  a  large 
estate;  A's  estate  is  worth  154760D.  ;  B's,  221784D.;  C's, 
341791D. :  A  has  contracted  debts  to  the  amount  of  47865D. ;  B, 
118464D.;  C,  187642D.;  required  the  amount  of  all  their 
debts,  and  what  they  will  all  be  worth  after  payment? 

Ans.  Debts,  353971 D.  ;  have  remaining,  364364D. 

69.  From  897640804235678  take  234796857912. 

70.  4  +  9  +  62  +  49  +  78423  +  64027=142574;  then~9876 
£=132698. 


SIMPLE  MULTIPLICATION. 

By  multiplication  we  can  perform  a  number  of  additions,  in 
a  shorter  and  more  easy  method,  or  it  is  a  number  repeated  a 
given  number  of  times  ;  as  5  multiplied  by  5  is  25  ;  so  if  five 
5's  be  added  together  it  would  be  the  same.  There  are  three 
parts  in  multiplication  which  require  particular  attention. 

1.  The  Multiplicand,  the  number  given  to  be  multiplied. 

2.  The  Multiplier,  the  (less)  number  by  which  you  multiply. 

3.  The  Product,  which  is  the  result,  or  sum  produced  by  the 
operation  of  multiplying,  and  is  just  as  many  times  larger  than 
the  multiplicand,  as  there  are  units  in  the  multiplier.  The  mul- 
tiplicand and  multiplier,  together,  are  sometimes  called y^c^or^. 

EXAMPLES,  FOR  MENTAL  EXERCISE. 

1.  What  will  2  pounds  of  coffee  cost,  at  14  cents  per  pound? 

2.  There  were  5  boys  who  gave  a  poor  man  3  cents  apiece ; 
how  many  cents  did  the  man  receive? 

3.  How  many  dollars  must  I  pay  for  12  yards  of  cloth,  that 
IS  worth  4  dollars  a  yard  ? 

4.  How  much  will  a  Jerseyman  get  for  25  melons,  if  he  sells 
them  at  5  cents  a  piece  ? 

5.  A  teacher  had  8  classes  in  his  school,  and  7  scholars  in 
each  class  ;  how  many  scholars  had  he  in  all  ? 

6.  There  are  24  hours  in  one  day ;  how  many  hours  in  3 
days  ? 

7.  If  1  pound  of  honey  is  worth  12  cents,  how  many  cents 
are  9  pounds  worth  1 

8.  14  and  what  number  make  20  ? 

9.  If  1  bushel  of  apples  cost  21  cents,  how  much  will  5  bush- 
els cost  ? 


SIMPLE    MULTIPLICATION. 


25 


MULTIPLICATION  TABLE. 


Twice 

5  times 

8  times 

11  times 

1  make  2 

1  make  5 

1  make  8 

1  make  11 

2      4 

2      10 

2      16 

2     22 

3      6 

3      15 

3      24 

3      33 

4      8 

4     20 

4      32 

4      44 

:>           10 

5     25 

5      40 

5      55 

6     12 

6      30 

6      48 

6      66 

7     14 

7      35 

7     56 

7      77 

8     16 

8      40 

8      64 

8      88 

9     18 

9     45 

9      72 

9      99 

10     20 

10      50 

10      80 

10     110 

U      22 

11      55 

11      88 

11     121 

12     24 

12      60 

12      96 

12     132 

3  times 

6  times 

9  times 

12  times 

1  make  3 

1  make  6 

1  make  9 

1  make  12 

2      6 

2     12 

2      18 

2     24 

3      9 

3      18 

3      27 

3      36 

4     12 

4     24 

4     36 

4     48 

5     15 

5     30 

5     45 

5      60 

6     18 

6     36 

6      54 

6      72 

7     21 

7     42 

7      63 

7      84 

8     24 

8      48 

8     72 

8      96 

9     27 

9      54 

9      81 

9     108 

10     30 

10      60 

10     90 

10     120 

11     33 

11     66 

11      99 

11     132 

12     36 

12      72 

12     108 

12  .    144 

4  times 

7  times 

10  times 

Twice 

1  make  4 

1  make  7 

1  make  10 

13  make  26 

2      8 

2     14 

2     20 

14     28 

3     12 

3     21 

3     30 

15     30 

4     16 

4     28 

4     40 

16     32 

6     20 

5      35 

5      50 

17     34 

6     24 

6     42 

6      60 

18     36 

7     28 

7     49 

7     70 

19     38 

8     32 

8     56 

8     80 

20     40 

9     36 

9      63 

9     90 

21      42 

10     40 

10     70 

10     100 

22     44 

11     44 

U     77 

U     110 

23     46 

1?     48 

12     84 

12     120 

24     48 

^  SIMPLE    iMULTIPLIGJLTlON. 

When  the  multiplier  does  not  exceed  12. 


1.  Set  down  the  multiplicand,  and  at  the  right  hana,  undei 
the  figure  or  figures,  of  the  multiplicand  set  the  multiplier. 

2.  Then  begin  with  the  imits,  and  multiply  all  the  figures  in 
the  multiplicand  in  succession,  and  set  down  their  several  prod- 
ucts, observing  to  carry  one  for  every  ten  to  the  product  of  the 
next  figure,  and  set  down  the  whole  of  the  last  product.  Proof, 
multiply  the  multiplier  by  the  multiplicand  ;  or  by  division. 


64032  multiplicand. 
5  X  multiplier. 


QUESTIONS. 

Explanation. — Begin  and  say  5  times 
2  are  10,  and  set  down  0 :  now  there 

is  1  to  carry  ;  then  say  5  times  3  are  ^ 

320160  product,  or  ans.  15,  and  1  is  16,  set  down  6  ;  now  there 
is  1  to  carry ;  5  times  0  is  0,  but  1  is 
is  1,  this  set  down  ;  now  you  have  none  to  carry,  because  it  is 
less  than  10  ;  then  5  times  4  are  20,  set  down  0  ;  now  you  have 
2  to  carry ;  then  5  times  6  are  30,  and  2  are  32 ;  set  down  the 
whole  number. 


1. 

12343 
2X 

2. 
345624 
2 

3. 

4532145 
3 

4. 
465832 
4 

5. 
1035678 
4 

24686 

691248 

13596435 

1863328 

4142712 

6. 
4789650 
5 

7. 
9650489 
5 

8. 
694763 
6 

9. 
141817614 
6 

10. 
95478654 
7 

11. 
143567 
8 

12. 
91456 
9 

13. 
234567 
10 

14. 
132454 
11 

15. 
134021 
12 

1148536 

823104 

2345670 

1456994 

1608252 

16.  What  cost  11  pounds  of  cheese  at  11  cents  per  pound? 

17.  What  would  you  give  for  17  quarts  of  corn  at  5  cents  per 
quart  ? 

18.  What  cost  24  quarts  of  salt,  at  9  cents  per  quart? 

19.  When  candles  are  worth  1^  cents  per  pound,  what  will 
11  pounds  cost? 


SIMPLE    MULTIPLICATION. 


27 


20.  What  will  34  pounds  of  nails  come  tx)  at  7 
pound? 

21.  How  much  will  47  quarts  of  chestnuts  cost  at  9  cents 
per  quart  ? 

22.  What  cost  11  dozen  of  eggs,  at  9  cents  per  dozen  ? 

23.  Multiply  120  by  2  ;  by  3. 

Ans.  i20x  2=240 

24.  Multiply  1211  by  5  ;   by  6. 

25.  Multiply  1211  by  7;  by  8. 
Multiply  65321  by  9 ;  by  6. 
Multiply  65321  by  8 ;  by  10. 


26. 
27. 
28. 
29. 
30. 
31. 


cents  per 


120x3=360+240=600. 

Ans.  13321. 

Ans.  18165. 

^71^.  979815. 

Ans.  1175778. 


Multiply  123456  by  11  ;  by  4. 
Multiply  123456  by  3  ;  by'5. 
Multiply  345612  by  3;  by  8. 
Multiply  345612  by  12  ;  by  7. 


Ans.   1851840. 

Ans.  987648. 

Ans.  3801732. 

Ans.  6566628. 

32.  Multiply  12345006789  by  3  ;  by  4.  Ans.  86415047523. 

33.  Multiply  12345006789  by  5 ;  by  6.  Ans.  135795074679. 
Multiply  9784076987  by  8 ;  by  12. 


34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42 
43. 
44. 


4218X 

7321 

87692 

95698 

10691 

31078 

109019 

900078 

278976 

12569769 


2  = 

3 

4 

5 

6 

7 

8 

9 
11 
12 


Ans.  8436. 

21963. 

350768. 

478490. 

64146. 

217546. 

872152. 

8100702. 

3068736. 

150837228. 


When  the  multiplier  exceeds  12,  and  consists  of  two  or  more 

figures. 

RULE. 

1 .  Set  down  the  multiplicand,  and  under  it  the  multiplier,  so 
that  units  may  stand  under  units,  tens  under  tens,  hundreds 
ander  hundreds,  &c.,  and  draw  a  line  under  them. 

2.  Multiply  each  figure  in  the  multiplicand  by  each  figure  in 
the  multiplier  separately,  beginning  at  the  right,  or  place  of 
units,  placing  the  result  directly  under  the  multiplier,  observing 
to  carry  1  for  every  10,  &c. ;  then  multiply  by  the  next  figure 
in  the  place  of  ten,  placing  the^r^^  figure  of  every  line  directly 
under  its  respective  multiplier. 

3.  After  multiplying  by  all  the  figures,  add  these  products  to- 
gether, and  their  amount  will  be  the  product  or  answer  required. 


2S 


SIMPLE    MULTIPLICATIOX. 


4.  When  ciphers  occur  at  the  right  hand  of  either  the  mul- 
tiplier or  multiplicand,  or  both,  omit  them  in  the  operation  and 
annex  them  to  the  product.  [Annex :  to  subjoin  at  the  end- 
right  hand.) 

QUESTIONS    AND    EXAMPLES. 


45. 

46. 

47. 

48 

49.     50.   51. 

14 

13 

28 

25 

35      30    42 

13x 

14 

25 

28 

30     35    22 

42 

52 

140 

200 

1050    150    84 

14 

13 

182 

56 
700 

50 
700 

10  X  90    84 

182 

10500  1050   924 

52. 

53. 

54. 

55.       56. 

7649 

145365402 

23567894 

356421345   3456784 

22 

14 

15 

23       24 

15298   581461608  117839470  1069264035  13827136 
15298   145365402   23567894   712842690   6913568 


16827 

8  2035115628 

353518410 

8197690935  82962816 

57. 

5234  X 

145. 

=        Ans.   758930 

58. 

129186 

98 

12660228 

59. 

23430 

230 

5388900 

60. 

5101 

300400 

1532340400 

61. 

674200 

62. 
7853214650 
462 

2104 

1418516800 

63. 
8653214578 
304* 

64. 

65. 

89700544315x4032 

6489784X2001 

66.  365420145321x3215: 

-~ 

Ans.  1174825767207015 

67.  10145034X2031  = 

Ans.  20604564054 

68.  Multiply  62123  by 

13. 

Ans.  807599 

•  IVIien  there  are  ciphers  between  the  significant  figures  of  the  multi- 
vlier,  we  may  omit  the  ciphers,  multiplying  by  the  significant  figures 
wtiy,  placing  the  first  figure  of  each  product  directly  under  the  figure  by 
which  we  multiply. 


SIMPLE    MULTIPLISATION. 


29 


69.  Multiply  35432  by  14 ;  by  15.  Ans.  1027528 

70.  Multiply  65217  by  16  ;  by  17.  2152161 

71.  Multiply  207812  by  19  ;  by  21.  8312480 

72.  207812      X25,  X35=  12468720 

73.  32100421        65,  85  4815063150 

74.  32100421        27,  33  1926025260 

75.  79814        29,  89 

76.  1978987        4809  9516948483 

77.  9807094        5047  49496403418 


When  the  multiplier  is  exactly  equal  to  any  two  figures  in  the 
multiplication  table. 

RULE. 

Multiply  first  by  one  of  those  figures,  and  that  product  by  the 
other,  and  the  last  product  will  be  the  answer. 

78.  79.  80.  81. 

521x16,  4X4  =  16,  or  thus,  521         4350x25     473213x35 
4  16  5  7 


2084 

3126   21750 

3312491 

4 

521       5 

5 

8336 

8336  108750 

16562455 

82. 

87698  X 

72  = 

Ans,  6314256 

83. 

75687 

56 

4238472 

84. 

34075 

36 

1226700 

85. 

47696 

144 

6868224 

86. 

23453 

81 

1899693 

87. 

45346 

49 

2221954 

88. 

74032 

64 

4738048 

89. 

85303 

100 

8530300 

90. 

63421 

110 

6976310 

91. 

79049 

81 

To  multiply  by  10, 

100, 

1000,  &;c.,  annex 

to  the  multiplicand 

all  the 

ciphers  in  the 

multiplier,  and  it  will  be  the  product  re- 

quired. 

92. 

421  X 

10= 

Ans.  4210 

93. 

4732 

100 

473200 

94. 

8619 

1000 

8619000 

95. 

18780 

10000 

187800000 

96.  John  was  employed  by  a  gentleman  67  days,  for  which 
he  received  18  cents  a  day;  how  many  cents  did  his  wages 
amount  to?  Ans.  1206. 

3* 


30  SIMPLE    MULTIPLICATION. 

97.  A  man  has  4  horses,  each  horse  has  4  shoes,  and  each 
shoe  has  8  nails ;  how  many  nails  are  there  in  all  their  shoes  ? 

Arts,  128. 

98.  If  32  quarts  make  1  bushel,  how  many  quarts  are  there 
in  ]  5  bushels  ?  Ans.  480. 

99.  If  112  pounds  make  one  hundred  weight  (gross),  how 
many  pounds  would  there  be  in  17  hundred  weight? 

Ans.  1904. 

100.  If  60  minutes  make  an  hour,  how  many  minutes  are 
there  in  1124  hours?  Ans.  67440. 

101.  If  a  man  shoidd  travel  247  days,  at  the  rate  of  38  miles 
a  day,  how  many  miles  would  he  travel  in  that  space  of  time  ? 

Ans.  9386, 

102.  If  a  company  of  98  men  should  receive  46D.  each,  as 
their  wages,  how  much  would  they  all  receive  ?     Ans.  4508D. 

103.  What  is  the  product  of  8505x64  ?  Ans.  544320. 

104.  What  sum  of  money  must  be  divided  among  500  men, 
so  that  each  man  may  receive  20D.  ?  Ans.  lOOOOD. 

105.  If  it  requires  5768  shingles  for  the  roof  of  a  house,  how 
many  would  it  require  for  16  houses  ?  Ans.  92288. 

106.  If  it  takes  3242  bricks  to  build  a  house,  how  many  will 
it  take  for  240  houses  ?  Ans.  778080. 

107.  There  are  4  houses,  each  house  has  22  windows,  and 
each  window  has  16  panes  of  glass  ;  how  many  panes  of  glass 
in  the  4  houses?  Ans.  1408. 

108.  In  a  certain  town  there  are  572  houses,  in  each  house 
6  rooms,  and  in  each  room  7  persons  ;  required  the  number  of 
persons.  Ans.  24024. 

109.  What  is  the  product  of  47865378X  13402  ? 

Ans.  641491795956. 

ADDITION    AND    MULTIPLICATION. 

110.  A  drover  purchased  84  oxen  for  54D.  each,  and  147 
cows  for  27D.  each  ;  how  many  cattle  had  he,  and  how  much 
did  they  cost?  Ans.  231  cattle,  cost  8505D. 

111.  Multiply  46  by  11  ;  71x18  ;  36x42  ;  add  their  sev- 
eral  products  together,  and  tell  their  amount.  Ans.  3296. 

112.  A  merchant  purchased  7  pieces  of  carpeting,  each  piece 
containing  27  yards;  and  4  pieces,  each  piece  containing  18 
yards  ;  how  many  yards  in  all  ?  Ans.  261. 

SUBTRACTION    AND    MULTIPLICATION. 

113.  Multiply  617  by  45,  from  that  product  subtract  6784. 

Ans.  20981. 


SIMPLE    MULTIPLICATION.  31 

114.  A  man  purchased  9  bags  of  salt,  each  bag  weighed  231 
pounds,  and  the  sacks  without  the  salt  weighed  46  pounds ; 
required  the  weight  of  salt  without  the  sacks.  Ans.  2033. 

115.  Multiply  47  by  16  ;  24  by  32  ;  from  their  sum  subtra.ct 
542.  Ans.  978. 

116.  If  there  be  16  ounces  in  one  pound,  how  many  ounces 
in  3  pounds  ?  in  5  pounds  ?  in  8  pounds  ?  in  1 1  pounds  ?  in 
129  pounds  ? 

117.  If  12  inches  make  1  foot,  how  many  inches  in  13  feet? 
in  29?  in  1700? 

118.  If  1  yard  of  riband  cost  42  cents,  how  many  cents  will 
17  yards  cost?  24  yards  ?  66  yards  ?  114  yards  ?  750  yards  ? 

119.  If  24  hours  make  1  day,  how  many  hours  in  4  days  ? 
in  1  week  ? 

120.  If  60  minutes  make  an  hour,  how  many  in  6  hours  ?  in 
96  hours  ?  in  789  hours  ? 

121.  At  42  cents  a  bushel,  what  will  315  bushels  of  corn 
cost?  Ati^.  13230. 

122.  If  63  gallons  make  a  hogshead,  how  many  gallons  in 
24  hogsheads  ?  in  38  ?  in  1974  ? 

123.  If  you  can  earn  QQ  cents  in  a  day,  how  much  in  97 
days  ?  in  159  ?  in  428  ? 

124.  If  the  velocity  of  a  car  on  a  railroad  is  18  miles  an 
hour,  required  the  number  of  miles  in  24  hours  ;  76  hours  ;  84 
hours ;  497  hours. 

REVIEW. 

What  is  multiplication  ?  What  is  the  number  to  be  multiplied 
called  ?  What  is  the  number  to  multiply  by  called  ?  What 
number  is  written  down  first  ?  How  must  the  multiplier  be  set 
under  the  multiplicand  ?  What  is  the  result  or  answer  called  ? 
How  many  times  larger  is  the  product  than  the  multiplicand  ? 
Where  do  you  begin  to  multiply  ?  Why  do  you  begin  to  multi- 
ply at  the  right  hand  figure  ?  How  do  you  place  the  first  product 
figure  of  each  line  ?  What  rule  does  multiplication  shorten  ? 
How  many  numbers  are  given  in  multiplication  ?  How  many 
numbers  are  required  ?  By  what  name  are  multiplier  and  mul- 
tiplicand together  called  ?  Repeat  the  rule  when  the  multiplier 
does  not  exceed  12.  When  the  multiplier  exceeds  12,  what 
is  the  rule  ?  How  do  you  prove  multiplication  ?  Repeat  the 
multiplication  table. 

125.  If  250  men  can  do  a  piece  of  work  in  218  days,  how 
many  days  will  it  require  for  1  man  to  do  the  same  ? 

Ans.  54500< 


32  SHORT    DIVISION. 

126.  What  will  100  bushels  of  corn  cost  at  95  cents  a  bush- 
el ?  Ans.  9500  cts. 

127.  What  will  1000  bushels  of  apples  cost  at  50  cents  a 
bushel  ?  Ans.  50000  cts. 

128.  Multiply  62123  by  13.  Arts,  807599 

129.  4078945X16=  ^n^.  65263120. 

130.  Multiply  9876543210  by  970845. 


SHORT  DIVISION. 

Division  teaches  to  divide  a  larger  number,  by  a  less,  into 
equal  parts,  and  is  a  short  method  of  performing  a  number  of 
subtractions  ;  thus,  if  we  wished  to  divide  25 D.  between  5  men, 
instead  of  subtracting  5  from  25  five  times,  we  would  divide  25 
by  5,  which  would  give  5D.  for  the  answer,  being  the  share  of 
each  man, because  5  is  contained  in  25  just  5  times, and  5  X  5 =25. 

There  are  four  parts  in  Division  that  require  attention,  name- 
ly :  Dividend,  Divisor,  Quotient,  and  Remainder. 

1 .  Dividend.  The  number  given  to  be  divided,  which  is  al- 
ways more  than  the  divisor. 

2.  Divisor.  The  number  given  by  which  the  dividend  is  to 
be  divided. 

3.  Quotient.  That  which  gives  the  number  of  times  the  divisor 
is  contained  in  the  dividend,  which  is  also  the  answer. 

4.  Remainder.  That  which  remains  after  all  the  figures  have 
been  brought  down  from  the  dividend,  and  the  last  subtraction 
performed  (if  any  remain)  will  be  of  the  same  denomination 
with  the  dividend,  and  always  less  than  the  divisor. 

When  the  divisor  does  not  exceed  12. 

RULE. 

1 .  Write  down  the  dividend,  and  draw  a  curve  line  at  the  left 
hand  side,  and  a  straight  line  under  the  dividend,  and  place  the 
divisor  at  the  left  hand  of  it ;  thus  :  Divisor  9)81  dividend. 

9  quotient. 

2.  Consider  how  many  times  the  divisor  is  contained  in  the 
first  figure  or  figures  of  the  dividend,  and  set  down  the  result, 
observing  whether  there  be  any  remainder,  and,  if  any,  carry  it 
to  the  left  of  the  next  figure,  and  consider  it  placed  there  as  so 
many  tens,  into  which  divide  as  before,  &c. ;  but  if  no  remainder, 
see  how  many  times  the  divisor  is  contained  in  the  next  figure. 

Proof. — Multiply  the  quotient  by  the  divisor,  and  add  in  the 
remainder,  if  any,  and  the  product  will  agree  with  the  dividend; 
if  the  operation  has  been  correctly  performed. 


SHORT    DIVISIOI^. 

33 

DIVISION  TABLE 

2  in  2      10  rem. 

4 

in  5 

1 

1 

6 

in  12 

2     0 

2        3      11 

4 

6 

1 

2 

7 

7 

1     0 

2        4      2     0 

4 

7 

1 

3 

7 

8 

1     1 

Z        5      2     1 

4 

8 

2 

0 

7 

9 

1     2 

2         6      3     0 

4 

9 

2 

1 

7 

10 

1     3 

2        7       3     1 

4 

10 

2 

2 

7 

11 

1     4 

2        8      4     0 

4 

11 

2 

3 

7 

12 

1     5 

2        9      4     1 

4 

12 

3 

0 

8 

8 

1     0 

2       10      5     0 

5 

5 

1 

0 

8 

9 

1      ] 

2       11       5     1 

5 

6 

1 

1 

8 

10 

1     2 

2      12      6     0 

5 

7 

1 

2 

8 

11 

1     3 

3        3       10 

5 

8 

1 

3 

8 

12 

1     4 

3        4      11 

5 

9 

1 

4 

9 

9 

1     0 

3        5       12 

5 

10 

2 

0 

9 

10 

1      1 

3        6      2     0 

5 

11 

2 

1 

9 

11 

1     2 

3        7      2     1 

5 

12 

2 

2 

9 

12 

1     3 

3        8      2     2 

6 

6 

1 

0 

10 

10 

1     0 

3         9      3     0          * 

6 

7 

1 

1 

10 

11 

1      1 

3       10      3     1 

6 

8 

1 

2 

10 

12 

1     2 

3       11       3     2 

6 

9 

1 

3 

11 

11 

1     0 

3       12      4     0 

6 

10 

1 

4 

11 

12 

1      1 

4        4       10 

6 

11 

1 

5 

12 

12 

1     0 

To  read  the  table, 

say  2  is  in  3 

once 

and  1 

over; 

3  in  5 

once  and  1  over,  or  re 

mainder. 

2  in  2      1 

4  in  4 

I 

6  in  6 

1 

8  in 

8 

1 

10  in 

10      1 

2       4     2 

4       8     J 

Z 

6 

12 

2 

8      1 

6 

2 

10 

20     2 

2       6      3 

4     12      , 

3 

6 

18 

3 

8     2 

4 

3 

10 

30      3 

2       8      4 

4     16      ^ 

i 

6 

24 

4 

8      3 

2 

4 

10 

40      4 

2     10      5 

4     20      , 

5 

6 

30 

5 

8     4 

0 

5 

10 

50      5 

2     12      6 

4     24 

5 

6 

36 

6 

8      4 

8 

6 

10 

60      6 

2     14      7 

4     28      ' 

7 

6 

42 

7 

8      5 

6 

7 

10 

70      7 

2     16      8 

4     32 

3 

6 

48 

8 

8      6 

4 

8 

10 

80      8 

2     18      9 

4     36 

9 

6 

54 

9 

8      7 

2 

9 

10 

90      9 

3       3      1 

5       5 

1 

7 

7 

1 

9 

9 

1 

11 

11      1 

3       6      2 

5     10 

2 

7 

14 

2 

9      1 

8 

2 

11 

22      2 

3       9      3 

5     15 

3 

7 

21 

3 

9      2 

7s 

3 

11 

33      3 

3     12      4 

5     20 

4 

7 

28 

4 

9      3 

6 

4 

11 

44     4 

3     15      5 

5     25 

5 

7 

35 

5 

9      4 

5 

5 

11 

55      5 

3     18      6 

5     30 

6 

7 

42 

6 

9      5 

4 

6 

11 

66      6 

3     21      7 

5     35 

7 

7 

49 

7 

9      6 

3 

7 

11 

77      7 

3     24      8 

5     40 

8 

7 

56 

8 

9      7 

2 

8 

11 

88      8 

3     27      9 

5     45 

9 

7 

63 

9 

9      8 

I 

9 

11 

99      9 

34  SHORT    DIVISION. 

28-r   7,  or  2^^  =:how  many  ?  49-t-  7,  or    Y=^ow"^^^y^ 

42-7-  6,  or  ^2_]2ow  many  ?  32-4-  4,  or    ^2_}jq^  niany  ? 

54-f-  9,  or  ^^^=howmany?  99-f-ll,or    ^|^=how  many  ? 

32^  8,  or  3^2_-how  many?  84-f-12,  or    f|-=:how  many ? 

33^11,  or  ff =how  many?  108-f-12,  or  i^^^nzhow  many? 

Note. — It  is  well  for  scholars  to  read  the  tables  in  class,  the 
same  as  reading  lessons,  by  which  means  they  will  learn  them 
in  a  few  hours.  The  most  of  the  Arithmetic  may  be  read  occa- 
sionally. 

How  many  oranges  could  you  buy  for  24  cents  at  6  cents  an 
orange  ? 

If  you  ride  4  miles  an  hour,  how  many  hours  would  it  take 
you  to  ride  36  miles  ? 

With  72  cents,  how  many  knives  could  you  buy  at  12  cents 
a-piece  ? 

How  many  times  9  in  81  ?  how  many  times  11  in  80,  and 
how  many  over  ? 

How  many  times  12  in  84  ?  how  many  times  12  in  136,  and 
how  many  over  ? 

QUESTIONS    AND    EXAMPLES. 

Divisor  3)976432  dividend.  Explanation. — Begin  and 

say  3  is  in  9  3  times,  which 

Quotient    325477 — 1  over,  or  rem.  set  down  ;    then  3  is  in  7 

X3  +  l  twice  and  1  over;  this  1  is 


now  supposed  to  be  placed 
Proof         976432  at  the  left  of  the  next  figure 

6,  which  would  be  16  ;  then 
say  3  is  in  16  5  times  and  1  over ;  then  3  in  14  4  times  and  2 
over ;  now  this  2  is  supposed  to  be  placed  at  the  left  of  the 
next  figure  3,  which  will  make  it  23  ;  then  3  is  in  23  7  times 
and  2  over ;  this  2  placed  at  the  left  of  the  next  figure  2  is  22  ; 
then  3  is  in  22  7  times  and  1  over,  a  remainder  which  must  be 
placed  at  the  right.  Then  multiply  the  quotient  by  the  divisor, 
and  add  in  the  1,  for  the  proof. 

JSlcte. — If  the  divisor  is  not  contained  in  the  next  figure  of  the 
dividend,  write  a  cipher  in  the  quotient,  and  join  this  figure  in 
the  dividend,  to  the  figure  next  to  it,  as  so  many  tens, 

1.  2.  3.  4.  5.  6. 

2)684        2)868       3)963       4)840       5)86421       6)784023 

342  ^434  321  210  17284—1     130670—3 


SHORT    DIV  SIGN. 


36 


7. 
7)94065832 

13437976 

11. 
11)2456321 


8.  9. 

8)4123045         9)6304532 


10. 
10)8465324 


15. 
4)9653450 


515380—5        700503—5 

12.  13. 

12)5840235  2)5678432 


223301—10 


486686—3 


16. 
5)8432105 


17. 
6)123465789 


846532—4 

14. 
3)6745076 


18. 
7)40345658 


19.  20.  21.  22. 

8)45673250        9)7847321         10)1213141516        11)84213567 


23. 
12)84213567 


24. 
12)64789057 


25. 
2)34789 


26. 
3)421562 


27.  240653-^4=  60163—1 

28.  321450-f-5==  64290—0 

29.  845354-7-6  =  140892—2 

30.  798543-^7=114077—4 

31.  643257-8=  80407—1 


32.  583460-f-  9=64828—8 

33.  146148-10=14614-8 

34.  367854-r- 11  =  33443— 3 

35.  845678-^12=70473—2 

36.  989789-^-100 


37.  A  farmer  sold  540  bushels  of  grain  to  9  men  ;  how  much 
did  each  man  receive  for  his  share  ?  Ans.  60  bush. 

38.  Divide  320D.  equally  among  5  persons.  Ans,  64D. 

39.  How  many  times  is  9  contained  in  6597?         Ans.  733. 

40.  If  a  man  spend  424D.  in  8  months,  how  much  is  that 
per  month?  Ans.  ^^D. 

41.  If  9  persons  sell  property  to  the  amount  of  20763D., 
how  much  will  each  man  receive  for  his  share  ?     Ans.  2307D. 

42.  Divide  378567  by  2 ;  by  3  :   2)378567  3)378567 


Ans.  315472 — 1  remainder. 

43.  Divide  278934  by  2,  by  3. 

44.  256788  by  3,  by  4. 

45.  256788  by  5,  by  6. 

46.  65342167by4,  by  5. 

47.  65342167by6,  by  7. 

48.  523467898  by  4,  by  6. 

49.  523467898  by  7,  by  8. 

50.  2653286  by  7,  by  8. 


189283—1  +  126189 

Ans.   232445. 

149793. 

94155— S 

29403974—5. 

20224956—3. 

218111623—6. 

140214615—4 

710700—12 


36  SHORT    DIVISION. 


What  is  Division  ?  How  many  parts  are  there  in  Division  f 
Which  is  the  Dividend  ?  Which  is  the  Divisor  ?  Which  is  the 
Quotient  ?  Which  is  the  Remainder  ?  When  the  Divisor  does 
not  exceed  12,  what  is  the  rule  ?  What  is  this  called  ?  How 
do  you  prove  Division  ?  Having  the  Dividend  and  Quotient 
given,  how  is  the  Divisor  found  ?  If  you  have  the  Divisor  and 
Quotient  given,  how  can  you  find  the  Dividend  ?  Recite  the 
Table. 

Note  to  Teachers. — It  is  expected  that  in  every  case  the  pu- 
pil will  commit  to  memory  the  tables  and  rules,  before  he  shall 
attempt  the  solution  of  the  questions.  It  is  evidently  erroneous 
for  a  scholar  to  attempt  to  work  by,  or  follow,  a  given  math- 
ematical rule,  unless  he  has  learned  it  and  understands  its  im- 
port, merely  from  a  given  example,  which  I  regret  to  say  is  too 
frequently  the  case  ;  consequently,  the  scholar  never  has  a  cor- 
rect knowledge  of  the  science  of  numbers.  The  questions,  or 
interrogatories,  given  in  the  Review,  can  be  used  by  the  teacher 
at  the  beginning  of  the  rule,  or  as  they  are  given,  or,  which 
would  be  still  better,  at  the  beginning  and  close  of  the  solution 
of  the  questions  in  the  rule. 


LONG  DIVISION. 

Long  Division  is  generally  used  when  the  divisor  is  more 
than  12. 

RULE. 

m 

1.  Write  down  the  dividend,  and  draw  curved  lines  at  the 
right  and  left  sides  of  the  dividend,  thus  :  )90( ,  and  place  the 
divisor  at  the  left  hand,  as  in  Short  Division. 

2.  See  how  often  the  divisor  is  contained  in  the  least  number 
of  figures  into  which  it  can  be  divided,  and  set  that  number  at 
the  right  hand  of  the  dividend,  which  is  the  multiplier. 

3.  Multiply  the  divisor  by  this  figure  in  the  quotient,  and 
place  the  result  under  the  figures  in  the  left  of  the  dividend, 
into  which  you  are  dividing. 

4.  Then  subtract  the  result  from  the  number  directly  over  it, 
and  set  down  the  remainder,  which  must  always  be  less  than 
the  divisor. 

5.  Bring  down  the  next  figure  of  the  dividend,  and  place  it 
at  the  right  of  the  remainder  5  if  this  number  is  less  than  the 
divisor,  place  a  cipher  in  the  quotient,  and  bring  down  another 


LONG    DIVISION. 


37 


{\fruYe  ;  which  forms  another  dividend,  into  which  divide  as  be- 
fore, and  so  continue  until  all  the  figures  are  biought  down  and 
divided. 

]\^ote. — If  there  be  ciphers  at  the  right  of  the  dividend  and 
divisor,  you  can  omit  an  equal  number  of  each,  by  placing  a 
period  at  the  left,  thus  :   1.00)10.00(10  answer. 


EXAMPLES    AND 

Dividend. 
Divisor  7)54306(7758  quotient. 
49 


53 
49 

40 
35 


QUESTIONS. 

Explanation. — You  can  not 
say  7  in  5,  because  5  is  less 
than  7 ;  but  say  7  in  54  7  times, 
7  times  7  being  49,  which  set 
down  under  54,  and  subtract, 
and  you  have  5  for  a  remain- 
der ;  then  bring  down  the  next 
figure,  which  is  3,  and  place  it 
to  the  right  of  the  5,  which  will 
make  53 — this  a  new  dividend , 
then  say  7  is  in  53  7  times 
and  4  remain ;  then  bring  down 
the  next  figure,  which  is  0  ; 
then  say  7  is  in  40,  5  times  7 
is  35  ;  set  this  down  under  40, 
then  subtract  35  from  40,  and  5  remain ;  then  bring  down  the 
next  figure  6  at  the  right  of  the  5,  which  will  make  56  ;  then  7 
is  in  56,  8  times  7  is  56,  and  0  remainder. 

1.  2.  3.  4. 

2)468(234         3)9636(3212         4)8456(2114         4)7324(1831 
4  2X         9  8  4 


56 
56 

0  remainder. 


6  468 
6 

proof 

6 
6 

4 
4 

33 
32 

8 
8 

3 
3 

5 

4 

12 
12 

6 
6 

16 
16 

4 
4 

5. 
9)87643(9738 

1] 

6. 

1)967884(87989 

7. 
12)138450(11537 

8. 
14)87631(6259 

9. 
15)161702(10780 

10. 
24)47653(1986 

LONG    DIVISION. 


11. 

36)87642(2434 

14. 
42)778614( 

17. 

72)840658( 

20. 
125)2486542( 

23. 
165)976405( 

26. 
5.00)10000.00( 


12. 
42)765431(18224 

15. 
55)460245( 

18. 
89)442356( 

21. 
139)5006789( 

24. 
185)372405( 
27. 
672)89746705( 


13. 
44)653145(14844 

16. 
64)128240( 

19. 
114)6743214( 

22. 

147)864221( 

25. 
200)5(>06030( 
28. 
842)1 0647896685( 


Divisor.  Dividend.  Quotient. 

29.  75  40231  536 

30.  422         253622         601 

31.  342       13699840        40058 

32.  3467        4586841         1323 

33.  1234       46447786        37640 

34.  1478   8769826000402    5933576454 

35.  1600       15463420        9664 

36.  27000       99607765        3689 

37.  How  many  times  is  176  contained  in  146524  ? 

Ans.  832,  92  rem. 

38.  How  many  times  is  250  contained  in  925500  ?     Ans.  3702. 

39.  Divide  5814D.  among  9  persons.  Ans,  646. 

40.  What  is  the  quotient  of  78656  divided  by  45  1 

Ans.  1747,  41  rem. 

41.  The  crew  of  a  ship  consisting  of  150  men,  are  entitled  to 
24750D.  prize  money  ;  what  is  the  share  of  each  man  1 

Ans.  165D. 

42.  What  number  must  be  multiplied  by  150  to  produce 
24750?  Ans,  165. 

43.  A  garrison  of  2441  men  have  received  9764  pounds  of 
flour  ;  how  much  is  the  share  of  each  man  ?        Ans.  4  pounds. 

44.  A  gentleman  bequeathed  his  estate,  which  was  estimated 
at  247655D.,  to  his  4  sons  and  only  daughter ;  how  much  did 
each  receive  ?  Ans.  49531 D 

45.  What  is  the  quotient  of  678976500  divided  by  6789  ? 

46.  What  is  the  quotient  of  10000000  divided  by  99  ? 

47.  Divide  500D.  equally  among  10  men;  40  men;  100 
men;  150  men. 


LONG    DIVI3I0X.  39 

48.  Divide  6780D.  among  15  men.  Ans.  452  each 

49.  Divide  975  pounds  of  flour  among  25  persons. 

Ans.  39  pounds  each. 

50.  Divide  100786  pounds  by  43.  Ans.  2343,  37  rem. 

51.  Divide  1600  bushels  of  wheat  among  40  men. 

Ans.  40  bush,  each 

52.  There  are  24  hours  in  a  day ;  how  many  days  are  there 
in  7248  hours?  ^n^.  302. 

53.  There  are  60  minutes  in  an  hour  ;  how  many  hours  in 
97680  minutes  ?  Ans,  1628  hours. 

54.  There  are  365  days  in  a  year ;  how  many  years  in  3285 
days  ?  Ans.  9  years. 

55.  There  are  63  gallons  in  a  hogshead  ;  how  many  hogsheads 
in  8796  gallons  ?  Ans.  139,  39  rem. 

56.  At  12  cents  per  pound,  how  many  pounds  can  you  have  for 
1728  cents  ?  Ans.  144  pounds. 

57.  If  the  dividend  is  4200,  and  divisor  48,  required  the 
quotient  ?  Ans,  87, 24  rem. 

58.  If  the  divisor  is  25,  and  the  dividend  5025,  what  is  the 
quotient?  -4^^.201. 

59.  If  you  sell  84  bushels  of  wheat  for  150  cents  a  bushel,  and 
take  your  pay  in  cloth  at  4D.  a  yard,  how  many  yards  will  you 
have?  Ans.  31^. 

60.  Divide  256976  by  41.  6267,  29  rem. 

61.  Divide  997816  by  59.  16912,  8  rem. 

62.  Divide  6283459  by  29.  216671. 

63.  Divide  37895429  by  112.  338352,5  rem. 

64.  Divide  29070  by  15 ;  by  18  (+  quot.)  3553. 

65.  Divide  29070  by  19;  by  17.  3240 

66.  Divide  10368  by  27;  by  36.  672. 

67.  Divide  10368  by  54;  by  18.  768. 

68.  Divide  2688  by  1 12 ;  by  224.  36. 

69.  Divide  101442075  by  4025.  25203. 

70.  Divide  978098745  by  78978. 


APPLICATION  OF  THE  PRECEDING  RULES. 

1.  If  you  add  476,  361,  842,  together,  and  divide  their  amount 
by  20,  what  number  will  result  ?  Ans.  83,  1 9  rem. 

2.  A.  has  2  farms,  one  of  732  acres,  the  other  241  acres  ;  if  he 
should  divide  the  land  equally  among  his  7  sons,  how  many  acres 
would  each  receive  ?  Ans.  139. 


40  APPLICATION    OF    THE    PRECEDING    RULES. 

3.  A  gentleman  has  2  sons  and  1  daughter;  he  is  worth 
697833D. ;  he  will  give  his  daughter  23261  ID.,  and  his  sons 
may  share  the  remainder  between  them  ;  was  the  estate  equally- 
divided  ? 

4.  If  you  multiply  1750  by  17,  and  divide  it  by  16,  what  will 
be  the  result?  Ans.  1859,  6  rem. 

5.  Divide  420D.  among  three  men  and  a  boy. 

6.  If  you  purchase  120  pounds  of  flour  for  7D.  and  sell  it  for 
6  cents  a  pound,  will  you  gain  or  lose,  and  how  much  ? 

7.  Make  out  a  bill  of  the  following  articles,  namely:  18 
pounds  of  nails  at  14  cents  per  pound  ;  47  pounds  of  sugar  at 
1 1  cents  per  pound ;  5  gallons  of  molasses  at  44  cents  per  gal- 
lon ;  4  pounds  of  tea  at  75  cents  per  pound  ;  9  yards  of  calico  at 
22  cents  per  yard  ;  and  2  brooms  at  25  cents  each. 

Ans.  15D.  37c. 

8.  A  gentleman  has  4  notes  of  hand :  the  first  is  for  474D. 
20c.  ;  the  second  for  760D.  42c. ;  the  third  for  285D.  68c. ;  the 
fourth  for  369D.  31c. ;  when  received,  he  must  pay  a  debt  of 
1562D.  96c.  ;  how  much  will  remain  ?  Ans.  326D.  65c, 

9.  "What  is  the  amount  of  40  bushels  of  corn  at  84  cents  a 
bushel ;  25  bushels  of  rye  at  97  cents  a  bushel ;  and  30  bushels 
of  wheat  at  ID.  25c.  a  bushel?  Ans.  95D.  35c. 

10.  What  is  the  quotient  of  55  multiplied  by  22  and  -7-15  ? 

Ans.  80,  10  rem 

11.  Purchased  1  set  of  chairs  for  4D.  25c. ;  table  7D.  50c. ; 
bureau  for  27D. ;  a  watch  for  60D.  50c. ;  writing-desk  for  18D. ; 
a  clock  for  40D. ;  required  the  amount.  Ans.  157D.  25c. 

12.  A  gentleman  of  Philadelphia  lately  purchased  4  tracts 
of  unimproved  land  in  the  western  country ;  in  the  first  tract 
there  are  2780  acres ;  in  the  second,  4027  acres ;  third,  3012 
acres  ;  fourth,  8760  acres,  all  of  which  is  worth  5D.  per  acre ; 
he  will  divide  all  the  lands  among  his  3  sons  ;  how  many  acres 
did  he  purchase,  and  how  much  land  will  each  son  receive,  and 
how  much  is  it  all  worth  ? 

Ans.  He  purchased  18579  acres  ;  each  son  will  receive  6193 
acres ;  worth  92895D. 

13.  A  farmer  has  3764  acres  of  land;  he  will  sell  A.  642 
acres,  B.  224  acres,  C.  180  acres,  and  D.  354  acres  ;  the  re- 
mainder is  to  be  equally  divided  among  his  4  children  ;  how 
much  land  wiU  each  receive,  and  how  much  will  each  share  be 
worth  at  65D.  an  acre  ?  Ans.  591  A. ;  worth  384 15D. 

14.  If  you  sell  44  bushels  of  oats  for  35c.  per  bushel,  25 
bushels  of  corn  for  76c.  per  bushel,  and  6  cords  of  wood  for 
4D.  per  cord,  how  much  would  you  receive  ?         Ans.  58D.  40c. 


;      APPLICATION    OF    THE    PRECEDING    RULES.  41 

15.  Required  the  amount  of  the  following  articles,  namely : 
1  set  of  knives  and  forks  at  3D. ;  1  Cashmere  shawl  at  750.^ 
47  yards  Irish  sheeting  at  90c.  per  yard ;  7  yards  of  broadcloth 
at  6D.  per  yard ;  one  Russian  hat  at  5D. ;  11  yards  of  silk  at 
2D.  per  yard  ?  Ans,  189D.  30c. 

16.  A  gentleman  deposited  in  bank,  at  one  time  4638D.,  at 
another  21 6D.,  at  another  8329D.,  at  another  1212D. ;  required 
the  amount  in  bank.  Ans,  14395D. 

17.  A  ship  in  sailing  to  a  distant  part  of  the  world  :  from  one 
port  to  another  was  6243  miles,  to  another  port  4123  miles,  to 
another  9401  miles,  and  thence  home  130  miles;  required  the 
number  of  miles  she  sailed.  Ans.  19897. 

18.  Supposing  you  gain  34568D.,  then  1245D.,  again  2467D., 
and  then  lose  2365D. ;  again,  you  gain  41210D.,  and  then  lose 
39300D. ;  how  much  will  you  have  left? 

19.  A  man  gains  367D.,then  loses  423D. ;  a  second  time  he 
gains  875D.,  and  loses  912D. ;  he  then  gains  1012D.  ;  how 
much  has  he  gained  in  all  ?  Ans.  91 9D. 

20.  A  farmer  agrees  to  furnish  a  merchant  40  bushels  of  rye 
at  60  cents  a  bushel,  and  take  his  pay  in  coffee  at  16  cents  a 
pound ;  how  much  coffee  will  he  receive  ?        Ans.  150  pounds. 

21.  A  farmer  sold  3  cows  at  25D.  each,  and  1  pair  of  oxen  at 
65D.,  he  agrees  to  take  in  payment  60  sheep ;  how  much  per 
head  do  his  sheep  cost  him?  Ans.  2D.  33c.  .3-f- 

22.  The  exports  of  the  United  States  from  October,  1841,  to 
October,  1842,  were  as  follows  :  of  products  of  the  sea,  2823010 
D. ;  of  the  forest,  5518262D. ;  of  agriculture,  73688113;  of 
manufactures,  1094061  ID.  The  total  value  of  the  imports  for 
the  same  period  was  100162087D.  How  much  did  the  total 
value  of  the  imports  exceed  that  of  the  exports  ? 

23.  If  M.  R.  is  worth  fourteen  millions,  two  hundred  and 
fifty  thousand  D.,  how  many  men  could  he  make  comfortably 
rich  by  giving  them  25  thousand  D.  each  ?  Ans.  570. 

24.  A  celebrated  personage  of  England  has  a  salary  of  seven 
hundred  and  fifty  thousand  dollars  annually ;  how  much  is  that 
daily,  and  how  many  teachers  would  it  pay  at  a  salary  of  500D. 
a  year?  -4w5.  daily  income  2054D.  79c.  4+  1500  teachers. 

25.  What  cost  6400  yards  of  riband  at  25c.  per  yard  ? 

Ans.  25rr:iD.  4)6400=1600D. 

26.  What  cost  3600  yds.  at  12^c.  a  yard?     12i=-iD.  8)3600. 

27.  What  cost  2400  pounds  of  cheese  at  6^0 .  a  pound  ? 

Ans.6l=j\D.  16)2400. 

28.  What  cost  600  bushels  of  potatoes  at  -J  of  a  dollar  a 
bushel?  Ans.  3)600=200D. 

4* 


43  APPLICATION    OF    THE    PRECEDING    RULES. 

29.  What  cost  50  bushels  of  wheat  at  ID.  25c.  a  bushel? 

Ans,  62D.  50c 

30.  Supposing  1800  apple-trees  to  be  planted  in  72  rows, 
how  many  trees  are  there  in  each  row  ?  Ans.  25. 

31.  A  merchant  bought  8200  barrels  of  flour;  he  then  sold 
3756  barrels  ;  he  again  bought  5000  barrels,  after  which  he  sold 
4879  barrels  ;  how  many  has  he  on  hand?  Ans.  4565. 

32.  A  man  sets  out  on  a  journey,  intending  to  travel  2450 
miles  ;  how  far  must  he  go  every  day  to  perform  the  journey  in 
50  days  ?  Ans.  49  miles. 

33.  The  quotient  of  an  operation  in  division  is  1763,  the 
dividend  8435955  ;  required  the  divisor.  Ans,  4785. 

34*  What  number  is  that  which,  being  multiplied  by  7969, 
the  product  will  be  1864746  ?  Afis.  234. 

REVIEW. 

What  is  Long  Division  ?  What  is  the  rule  ?  At  which  hand 
of  the  dividend  must  the  divisor  be  written  ?  Why  do  you  be- 
gin at  the  left  hand  of  the  dividend  to  divide  ?  Ans.  Because 
numbers  decrease,  &c.  Under  this  rule,  where  must  the  quo- 
tient be  written  ?  How  many  figures  of  the  dividend  must  first 
be  taken  ?  How  can  you  find  the  one  half  of  any  number  ?  one 
fourth  ?  one  sixth  ?  one  eighth  ?  &c.  How  many  rules  have 
you  now  been  through  ?  Name  them  ?  Why  are  they  called 
the  principal  or  fundamental  rules  of  the  science  ?  Ans.  Be- 
cause they  are  the  foundation  of  all  the  other  rules,  and  by  their 
use  and  operation  all  calculations  in  arithmetic  are  performed. 
Perform  the  following  examples  on  the  slate,  as  the  signs  indi- 
cate :  878344-284+65  +  32  +  100=88315  Ans.  876345723 
—267001345  =  609344378  Ans.  692784578  X  27839421  rzr 
19286721529249338  Ans.  202884150-^4025  =  50406  Ans. 
2600—600=2000  +  1828  =  3828  Ans.  9788x97^29x17  + 
79—400x92. 

35.  From  95,000,000  take  18,999,999;  from  777,777-- 
688,888. 

36.  If  a  miller  should  purchase  578  bushels  of  wheat  for 
482D.,  and  sell  482  bushels  for  375D.,  how  many  bushels  will 
he  have  left,  and  what  sum  will  he  have  paid  for  the  quantity 
remaining?  Ans.  96  bushels;  paid  107D. 

37.  From  eleven  millions  take  eleven  hundred  and  eleven. 

Ans.  10998889, 

38.  What  is  the  difference  between  84  and  76  ?  90  and  22  ! 
400  and  380  ?  7868  and  9897  ? 


DECIMAL    FRACTIONS.  43 

Methuselah  died  aged  969  years,  and  Adam  aged  930  ;  what 
is  the  difference  of  their  ages  ?  How  many  years  were  required 
to  have  extended  the  life  of  Methuselah  to  1000  ?  How  many 
for  Adam  1  How  many  years  are  their  united  ages  ?  How 
many  years  since  the  death  of  Adam?  How  long  from  the 
death  of  Adam  to  the  deluge  ?  How  many  years  from  the  del- 
uge to  the  present  time  ?  In  what  year  was  America  discov- 
ered by  Columbus  ?  How  many  years  since  ?  How  many 
years  since  the  declaration  of  independence  1  How  are  you 
pleased  with  the  science  of  Arithmetic  ? 


DECIMAL  FRACTIONS, 

Adapted  to  the  Currency  of  the  United  States. 

We  now  come  to  treat  of  Decimals  in  connexion  with  the 
currency  of  this  country,  by  which  it  is  believed  that  calcula- 
tions generally,  at  least  where  money  is  concerned,  will  be  much 
facilitated,  and  bring  to  view  a  better  and  easier  method  of  com- 
putation than  has  heretofore  been  in  general  use.  That  fractions 
must  of  necessity  occur  in  almost  every  transaction,  none  will 
presume  to  deny,  and  all  will  admit  that  no  part  of  the  science 
of  Arithmetic  is  more  difficult  of  comprehension  to  the  juvenile 
mind  than  fractions,  especially  those  termed  vulgar ;  but  in  this 
treatise,  this  part  of  Arithmetic  is  introduced  in  a  pleasing  and 
agreeable  manner,  particularly  calculated  to  attract  the  attention 
of  youth,  and  overcome  every  obstacle  and  difficulty ;  and  the 
author  entertains  the  belief  that  he  has  in  part  succeeded. 

Decimal  is  derived  from  the  Latin  word  decern  (signifying 
ten,  because  they  increase  and  decrease  in  a  tenfold  proportion, 
like  whole  numbcBs),  and  is  a  part  of  a  whole  number,  or  unit, 
which  is  distinguished  by  a  period,  decimal  point,  or  separatrix, 
placed  at  the  left  of  the  figure  or  figures,  thus:  .5=:y^Q-,  or  one 
half;  .25 =y2_5^= twenty-five  hundredths = one  quarter ;  6  .8,  six 
and  eight  tenths;  .15 =y^^5__ fifteen  hundredths;  .144=:j^qYo» 
one  hundred  and  forty-four  thousandths,  &c. ;  the  number  above 
the  line,  ^^"^j  is  called  the  numerator,  and  the  number  below  the 
line,  Yooo?  ^^  called  the  denominator,  and  must  consist  of  one 
place  more  than  the  numerator,  and  shows  the  number  of  j:|^rta 
into  which  a  unit  or  anything  is  divided,  thus,  ^Vo  '  ^^  7^^  ^^^ 
25  parts  in  100  parts,  you  own  one  fourth,  because  25x4  =  100== 
4"§§,  the  numerator  and  denominator  being  alike,  are  equal  to  1 
The  word  fraction  implies  broken  parts  of  a  unit  or  whole  number 


44  DECIMAL    FRACTrONS. 

In  counting  decimals  from  left  to  right,  they  decrease  in  a  tenfold 
proportion,  with  a  view  to  give  the  decimal  expression  ;  but  to 
enumerate  them  from  right  to  left,  they  increase  in  a  tenfold  pro- 
portion, the  same  as  whole  numbers.  Ciphers  annexed  to  a  deci- 
mal do  not  increase  its  value,  as  in  whole  numbers  ;  their  value  is 
determined  by  their  distance  from  the  units'  place,  or  decimal 
point;  thus  .1,  or  .10,  are  equal,  being  read  one  tenth  and  ten 
hundredths,  which  is  the  same ;  but  when  prefixed,  that  is,  placed 
at  the  left  of  a  figure,  it  decreases  in  a  tenfold  proportion,  thus, 
.1  .01  ;  in  this  position  their  value  is  different ;  figure  1  and  0 
having  changed  places,  it  has  decreased  the  value  of  1  in  a 
tenfold  proportion,  it  being  only  one  tenth  part  as  much  as  figure 
.1  ;  .5,  .05,  .005=-3-^o,  j^,  j^^,  .75=1=/^%,  &c. 

A  mixed  number  is  that  which  consists  of  a  whole  number 
and  decimal,  as  4.4,  21.42,  100.75,  &;c. 

There  are  various  kinds  of  decimals,  such  as  circulating 
decimals  ;  but  the  limits  of  this  work  will  not  permit  a  full  in- 
vestigation ;  neither  does  the  occasion  require  it,  suffice  to  say ; 
in  circulating  decimals,  if  one  figure  only  repeats,  it  is  called  a 
single  repetend,  as  .1111,  .6666,  &;c. 

A  compound  repetend,  thus  :  .0101,  .379379,  &;c.  There 
are  a  variety  of  examples  under  this  head,  but  in  most  cases 
three  or  four  places  of  decimals  will  be  sufficient,  unless  great 
accuracy  is  required.  "We  will  now  introduce  decimals  as  ap- 
plicable to  the  currency  of  this  country,  which  in  its  nature  and 
operations  is  purely  decimal.  The  laws  of  our  country  require 
that  all  transactions  in  money,  both  as  relates  to  government  and 
individuals,  shall  be  performed  in  dollars  and  cents,  or,  as  it  is 
termed,  "federal  currency."  This  currency  increases  and 
decreases  like  decimals,  in  a  tenfold  proportion,  thus  : — 

10  mills  equal  1  cent,  is  -         10  mills,     m.  mill. 

10  cents      "     1  dime    -         -       100     "         d.  dime. 

10  dimes    ''     1  dollar  -         -     1000     "         D.  dollar. 

10  dollars"  1  eagle  -  -10000  "  E.  eagle. 
It  is  customary  with  accountants  to  use  only  two  of  the  above 
denominations,  namely,  Dollars  and  Cents,  the  cents  being 
hundredths  of  a  dollar  ;  and  the  fractional  part  of  a  cent  is  ex- 
pressed thus :  i,  one  quarter;  -i,  one  half;  |,  three  quarters  of 
a  cent ;  but  it  would  be  more  correct  to  give  the  exact  number 
of  mills.  There  is  but  one  difficulty  in  learning  decimals — that 
is,  where  to  place  the  decimal  point  or  separatrix,  which  will 
depend  entirely  upon  the  nature  of  the  question,  for  which  there 
is  a  sufficient  number  of  rules  and  examples  ;  and  in  every  other 
respect,  their  operation  is  the  same  as  whole  numbers. 


DECIMAL    FRACTIONS.  45 


AMERICAN    COINS. 


The  Mill  is  not  a  coin,  but  the  tenth  part  of  a  cent. 

The  Half-Cent  is  a  copper  coin,  200  being  equal  to  one  dol- 
lar. 

The  Cent  is  a  copper  coin,  TOO  being  equal  to  one  dollar. 

The  Half-Dime  is  the  smallest  silver  coin,  being  equal  in 
value  to  5  cents,  or  20  to  one  dollar. 

The  Dime  is  a  silver  coin,  being  equal  to  10  cents,  or  10  to 
one  dollar. 

The  Quarter-Dollar  is  a  silver  coin,  =25  cents,  or  4  to  a  dollar. 

The  Half-Dollar  is  a  silver  coin,  =50  cents,  or  2  to  a  dollar. 

The  Dollar  is  the  largest  silver  coin,  equal  to  100  cents,  -^^ 
of  an  Eagle. 

The  Quarter-Eagle  is  the  smallest  gold  coin,  =2D.  50cts. 

The  Half-Eagle  is  a  gold  coin,  =5D. 

The  Eagle  is  the  largest  gold  coin.  =10D. 

The  above  coins,  in  their  composition,  are  not  of  pure  gold 
and  silver,  but  composed  in  part  of  alloj/,  a  table  of  which  will 
be  given  in  its  proper  place. 

To  write  sums  in  dollars  and  cents,  the  dollars  are  placed  at 
the  left,  and  the  cents  at  the  right  hand,  separated  by  a  period. 

D.     c.  D.     c.  D.    c. 

Thus,   24  .25.     or   20  .09:     or  18  .00:    (see  the  following 
table.) 


:=:  ^  r^      ^  S 

;©;=!  o      ^  V- 

•no            o  c 

s  of 
ofd 
tens 

tens 
ens  i 

TABLE    I. 

A 

ee 

S-gJs  Ss 

B 

o  c  bD:=i  s  c 

o 

^   P   rt   o  .5   O 
EhJUhqqo 

D. 

c. 

parts. 

c. 

1   .23 

1 

and  .23 

or     123 

k        1  3  .12 

13 

.12 

1312 

r        4  2  .09 

42 

.09 

4209 

6  7  2  .18 

672 

.18 

67218 

5  6  9  .04|- 

569 

.04 

and    f- 

56904 

.75 

7  0  15  .42^ 

7015 

.42 

t 

701542 

.50 

2  14  6  .81A 

2146 

.81 

214681 

.25 

To  reduce  oents  to  dollars,  divide  by  100,  or  separate  two 
figures  from  the  right,  and  those  on  the  left  will  be  dollars  ;  to 


Too 
Tooo 


46  DECIMAL    FRACTIONS. 

change  fourths  of  a  cent  to  cents,  divide  by  4  ;  to  change  halves 
of  a  cent  to  cents,  divide  by  2  ;  to  change  thirds  of  a  cent  to 
cents,  divide  by  3  ;  to  change  mills  to  dollars,  divide  by  1000, 
or  separate  three  figures  from  the  right  of  their  number,  thus : 
1.000=1D. 

TABLE    II. 

.  o  g  «  -5 

TO    »    o      ^'^  ^  to 

c«    P!    «>    rt    c    03      .  ^ 

2  g  S  S  2  ^  ^  ^^^  'S  ^ 

y5_^                r=                .5  «              5  tenths. 

-J^i              =.0  7  '*              7  hund's. 

=               .0  4  5  "45  thous's. 

illio         =                .14  2  5  "        1425  tenth's 

6J^               =           6  .7  «      6  &  7  tenths. 

4iWo75(r(5   =           4  .0  0  0  0  0  7           "    4  &  7  mill's. 

22j2^%             =         22  .2  4  "  22  &  24  hun'ths. 

3/00-0^0-000   =            3  .0  0  0  0  0  0  4       "    3  &  4  10  mill's 

444  4  4  4  .0  0  0  0  0  0  0       "  444  4  h.  &  44 

TABLE    III. 

Whole  numbers.  Decimals. 

•  nd   CO  *J   '-'   S 

"s  3     ^  §   .  ^-  ^  <=^     g.'^  1 .0 

0«4-ic!<^'^Sl5  5         ^    ?^    3    C  =^  "S    S 

owj=!bJ»gflg.-i^  rSoo^onH^oo 

oo-?oo_2pgj3  00000 -^^  000 

987654321  .123456789 

QUESTIONS    AND    ANSWERS. 

Quest.  When  the  folio w^ing  fractions  occur,  such  as  -f^,  -f^^^ 
yJI^,  how  is  a  unit  supposed  to  be  divided  ?  Ans.  Into  10  equal 
parts,  called  tenths ;  and  each  tenth  into  10  other  equal  parts, 
called  hundredths;    and  each  hundredth  into  10  more  equal 


ADDITION    OF    DECIMALS.  47 

parts,  called  thousandths,  &c.  Quest.  How  will  you  write  them 
down  so  as  to  give  the  decimal  expression  ?  Ans.  By  taking 
away  the  denominator,  and  placing  a  period,  or  decimal  point, 
before  the  numerator,  thus :  .5,  .25,  .75,  .8,  .2,  &c.  Quest, 
What  is  the  use  of  a  decimal  point  in  fractions  ?  Ans.  It  shows 
the  place  of  the  units,  and  separates  the  fraction  from  the  figures 
at  the  left  (if  any)  which  are  whole  numbers,  and  may  be  called 
the  separatrix.  Quest.  Write  down  y^^  (7  hundredths)  decimally. 
Ans.  .07,  also  yoVo'=-^^^  ^^^  T5W  =  •004= eight  thousandths 
and  four  thousandths.  Quest.  How  do  decimals  decrease  in 
value  from  left  to  right?  Quest.  How  do  you  determine  the 
value  of  a  decimal  ?  Ans.  By  its  distance  from  the  decimal  point. 
^  For  a  more  full  and  perfect  explanation  of  fractions,  see  Vul- 
gar Fractions,  which,  if  the  teacher  please,  can  follow  decimals. 


ADDITION  OF  DECIMALS. 

RULE. 

1 .  Write  down  the  numbers  under  each  other,  observing  to 
place  tenths  under  tenths,  hundredths  under  hundreths,  &c 
Be  particular  that  the  decimal  points  stand  directly  under  each 
other  in  a  perpendicular  line,  both  in  the  given  number  and  in 
the  sum.  2.  Then  perform  the  operation  the  same  as  in  Sim- 
ple Addition.     Proo/— as  Simple  Addition. 

EXAMPLES    AND    QUESTIONS. 

I.      2.       3.  4.  5.      6. 

41  .653  46  .23456  12  .3456  48  .9108  21  .037  49  .607 

36  .05  24  .90400  7  .891  1  .8191  15  .122  50  .421 

24  .009  17  .00411  2  .34  3  .1030  12  .042  18  .1610 

1  .6    3  .01111  5  .6  .7012  10  .120  71  .65843 


103  .312  91  .15378  28  .1766  54  .5341  58  .321 

7.  Add  1?8  .34565+7  .891+2  .34  +  14  .0011  together. 

^71^.36  57775. 

8.  Add  .7509+  0074+.69+.8408+.6109  together. 

Ans.  2  .9000. 

9.  Add  .7569  +  . 25+, 654+.199  together.         Ans.  1  .8599 


48  SUBTRACTION    OF    DECIMALS. 

10.  Add  71   .467+27    .94+16    .084+98  .009+86  .5   to- 
gether. Ans.  300  .000 

11.  Add  to  9  .999999  the  millionth  part  of  a  unit. 

Ans.  10    000000 

12.  13.  14.  15. 

D.    c.  m.  D.  d.  c.  m.  D.    d.  c.  m.  E.    D.  d.  c.  ra.dec.of  m 

14  .25  .6  12  .2  .6  .4  147  .6  .3  .4  121  .7  .6  .5  .3  .25 
13  .14  .4  10  .8  .7  .3  121  .7  .5  .4  324  .6  .5  .4  .5  .75 
13  .12  .3  15  .7  .6  .4  134  .3  .4  .6  242  .5  .6  .0  ,4  .25 

15  .10  .2  16  .4  .3  .2  147  .6  .5  .4  221  .3  .4  .5  .G  .50 


55  .62  .5  55  .3  .3  .3  551  .3  .8  .8  910  .3  .2  .5  .9  .75 

16.  Add  278D.  9d.  7c.  4m.  .21  +  87D.  6d.  9c.  5ni.  .75  + 
396D.  4d.  7c.  6m.  .25  +  464D.  6d.  3c.  5m.  .35,  together. 

Ans.  1227D.  7d.  8c.  Im.  .56. 

17.  A  gentleman  has  the  following  charges  entered  in  his 
ledger:  A.  420D.  27c.  5m.;  B.  671D.  87c.  6m.;  C.  742D. 
33c.  4m.  ;  D.  621D.  25c.  7m.;  E.  520D.  47c.  6m. ;  required 
the  amount.  Ans.  2976D.  21c.  8m. 

18.  A  farmer  sold  his  wheat  for  1724D.  87c.  5m.  ;  rye, 
1296D.  18c.  .75  ;  corn,  964D.  25c.  8m. ;  oats,  250D.  37c.  5m. ; 
required  the  amount  of  the  value  of  his  produce. 

Ans.  4235D.  69c.  .55. 

19.  A  drover  paid  for  his  drove  as  follows:  oxen,  1800D. 
75c.;  cows,  878D.  50c.;  sheep,  502D.  31c. ;  he  also  purchased 
2  horses  for  75D.  50c.  each ;  required  the  cost  of  his  drove. 

Ans.  3332D.  56c. 

20.  Add  420E.  8D.  27c.  6m.  .75  +  371E.  8D.  29c.  4m.  .25 
+781E.  5D.  17c.  .8m.  37+416E.  7D.  14c.  5m.  .50,  together. 

Ans.  1990E.  8D.  89c.  4m.  .87 


SUBTRACTION  OF  DECIMALS. 


RULE. 


1.  Write  down  the  numbers  the  same  as  in  Simple  Subtrac- 
tion, observing  that  the  decimal  points  stand  directly  under  each 
other.  2.  Then  subtract  in  the  same  way  as  in  whole  numbers 
and  place  the  decimal  point  in  the  remainder  under  those  above 
it.     Proof— 0.8  Simple  Subtraction. 


SUBTRACTION    OF    DECIMALS. 


41» 


From   125  .64000 
Take    95  .58756 

145  .00  14 
96  .84   5 

.674 
.91 

2  .764 
2  .371 

761  .8109 
18  .9113 

Rem. 

30  .05244 

48  .16   8 

.764 

:  0  .393 

742  .899« 

6, 
271  .0 
215  .7 

7, 
464  .000 
376  .784 

8. 

719  .10009 

7  .121 

9. 
270  .200 
75  .4075 

10. 
480  .OOOCi 
245  .0075 

11.         12. 
107  .0000    236  .OOC 
.0007       .54S 

) 

1 

13. 

1000000 
.1 

D. 

%  431 

271 

14. 
c.  m. 
.76  .8 
.25  .6 

15. 
D.  c.  m. 
671  .37  .4 
581  .65  .3 

16. 
E.  D.  d 

331  .7  .8 
224  .8  .5 

.  c.  m. 
.5  .4 
.6  .2 

160 

.51  .2 

89  .72  .1 

106  .9  .2 

;  .9  .2 

17. 
E.  D.  d.  c,  m.  dec. 
336  .4  .0  .2  .4  .25 
237  .8  .1  .3  .2  .10 

18. 

496  .0784 
379  .6809 

19. 

700  .94870 
199  .48397 

20. 

42  .84 
19  .96432 

21.         22.         23. 

658  .00000  66666  .555556  78888  . 

1  .77777  57777  ,666666     69999  . 

7777  1  J 
8888    J 

24. 
000000009 
300000011 

25.  From  1078D.  7d.  8c.,  take  984D.  4d.  9c. 

Ans.  94D.  2d.  9c. 

26.  From  99D.  99c.  9m.,  take  9d.  9m.  Ans.  99D.  09c. 

27.  From  19E.,  take  ID.  99c.  .5.  Ans.  188D.  00c.  5m. 

28.  A  merchant  deposited  in  bank  900E.,  he  drew  out  at  one 
time  2434D.  50c.  5  at  another  3224D.  18c.  .75  ;  how  much  has 
he  in  bank  ? 

29.  A.  sold  wheat  to  B.  to  the  amount  of  7840.  50c.  ;  corn, 
347D.  75c.  ;  B.  paid  in  part  for  the  wheat,  525D.  25c. ;  for 
the  corn,  235D.  12c.  5m. ;  how  much  remains  due  from  B.  to 
A.?  Ans.37]I).S7c.5m, 

30.  A  merchant  in  the  city  sent  his  clerk  into  the  country  to 
pollect  debts  ;  of  A.  be  received  475D,  65c.  ;  of  B.  8740.  87o 


M)  MULTIPLICATION    OF    DECIMALS. 

5m.  ;  of  C.  721D.  35c.  5m.  ;  on  his  return  he  lost  565D.  ^H/^ 
5m. ;  how  much  will  he  be  able  to  pay  his  employer?  "^  j 

Ans,  1506D.  50c.  5m. 

31.  A.  paid  at  the  store,  for  flax  12D.  25c.,  tallow  13D. 
27c.  5m.,  butter  14D.  85c.,  groceries  18D.  25c.  4m.,  for  which 
he  gave  the  merchant  a  bank-note  of  lOOD. ;  how  much  change 
must  A.  receive  back?  Ans.  41D.  37c.  Im. 

32.  What  is  the  difference  between  900d.  and  900c.  ? 

Ans.  SID. 

33.  A  man,  on  leaving  home,  left  in  his  desk  lOOOOD.;  during 
his  absence,  his  family  had  occasion  to  use  a  part  of  it,  but  could 
not  tell  how  much ;  on  counting  it,  he  found  that  he  had  7321D. 
47c.  6Tn. ;  how  much  was  taken?  Ans.  2678D.  52c.  4m. 

34.  From  799999  .8888888,  take  654321   .9999999. 


MULTIPLICATION  OF  DECIMALS. 

RULE. 

1.  Write  down  the  multiplicand,  and  under  it  the  multiplier, 
in  the  same  manner  as  Simple  MuUiplication,  and  multiply 
without  regard  to  decimal  points. 

2.  When  you  have  finished  multiplying,  begin  at  the  right- 
hand  figure  of  the  product,  and  count  off  as  many  figures  toward 
the  left  as  there  are  decimal  figures  in  the  multiplicand  and 
multiplier,  and  there  place  the  decimal  point.  If  the  number 
of  figures  in  the  product  be  less  than  the  decimal  figures  in  the 
multiplicand  and  multiplier,  prefix  a  suflficient  number  of  ciphers 
at  the  left  of  the  product,  to  equal  those  above  the  line,  then 
place  the  decimal  point  at  the  left  of  the  ciphers. 

EXAMPLES    AND    QUESTIONS. 


1. 

D.3.024 
D.2.023X 

2. 

Feet  25.238 
12.17 

3. 

.007853 
.035 

9072 
6048 
6048 

176666 
25238 
50476 
25238 

39265 
23559 

000274855 

D.6  117552 

F.307,14646 

MULTIPLICATIOJf    0F    DKCIMALg.  51 

4.  5.  6. 

.004  Yards  6.21  D.7.02 

.004  D.2.25  5.27 


R  .000016  3105  4914 

Multiply  4  by  4=16,  then  1242  1404 

ring  down  the  4  ciphers  and         1242  3510 

make  the  point ;  or  thus :  .  ,  ,  4    

X  .  ,  ,  4  (four  thousandths)^     D13.9725  D36.99.5.4 

16;  prefix  4  ciphers,  .000016. 

Explanation. — In  table  2,  the  pupil  will  notice  that  ID.  is  the 
unit  in  our  currency,  and  that  all  of  the  less  denominations  are 
so  many  decimals,  or  fractional  parts  of  the  dollar,  decreasing 
in  a  tenfold  proportion  from  the  decimal  point,  as  dimes,  cents, 
mills,  &c. ;  therefore,  in  the  (1)  example,  the  multiplicand  is 
D3.02  cents  and  4  mills,  or  y^^^  of  a  dollar.  After  you  have 
multipled  byall  the  figures  in  the  multiplier,  that  is,  D2.02  cts. 
and  3  mills,  or  yffo"  ^^  ^  dollar,  you  count  off  as  many  decimal 
figures  in  the  product  as  you  have  in  ±e  multiplicand  and  mul- 
tiplier, namely  6,  and  there  make  the  decimal  point,  and  the  figure 
or  figures  at  the  left  of  the  point  will  be  dollars=6D.  lie.  7m. 
and  live  hundred  and  fifty*two  thousandths  of  a  mill,  which  is  a 
little  more  than  half  a  mill,  &;c.  In  the  (2)  example  25.238  feet 
are  given  to  be  multiplied  by  12.17  feet ;  multiply  and  count  off 
as  before,  and  you  have  five  decimal  places,  and  three  integers, 
or  whole  numbers,  which  is  307  feet ;  .14646  of  a  foot,  equal  to 
about  1|-  inches.  In  the  (3)  example,  after  multiplying,  we  find 
the  product  to  consist  only  of  6  figures  ;  therefore  we  must  prefix 

3  ciphers  to  make  the  number  equal  to  the  multiplicand  and 
multipler.  In  the  (4)  example,  the  product  is  even,  or  equal  to 
the  multiplicand  and  multiplier.  In  the  (5)  example,  we  multi- 
ply 6.21  yards  by  D2.25,  this  will  give  the  value  of  the  cloth  at 
that  price,  there  are  4  given  decimal  figures,  therefore  count  off 

4  in  the  product,  and  we  have  Dl 3.9725  =  13  dollars  97  cents 
and  one  quarter  of  a  cent.  In  the  (6)  example  we  multiply  dollars 
by  dollars,  or  dollars  and  fractions,  and  the  product  may  be  point- 
ed off  thus  :  D.36.9954,  or  36D.  99c.  5.4m. ;  observe  that  the 
cents  must  always  have  two  places,  thus  :  (75)  unless  you  divide 
it  into  dimes  and  cents,  then  but  one  place,  as,  7d.  5c.,  which 
is  the  same  =r 75  cents.  (As  a  fraction,  or  decimal,  is  always 
less  than  one,  or  unity,  it  follows  that  unity  can  not  be  produced 
by  the  multiplication  of  decimals,  for  .9x.9  =  .81;  or  .999 X 
.999  =  .998001,  (fee;  but  whole  numbers  maybe  produced  by 
the  addition  of  several  fractions,  thus  :  .5  +  .5-|-.54-.5=2.0 ;  or 
^+Tiy+l^TJ+T^=T^=2  '  ^^^  ^  fraction  may  be  divided  by  a 


63  MULTIPLICATION    OF    DECIMALS. 

whole  number,  whether  it  be  a  decimal  or  vulgar  ;  but  the  quo- 
tient must  decrease  from  the  decimal  point  in  proportion  to 
the  value  of  the  divisor,  thus :  .624672-^482,  v/hole  number, 
=  .001296  Ans, 

7.  What  cost  27.5  pounds  of  butter  at  12.5  cents  per  pound? 
27.5  pounds 

.125  X  Explanation. — First  multiply  in  the  usual  way 

' by  the  multiplier,  which  is  y\fo\  of  a  dollar:^ 

1375  12^  cents,  or  -i   of  a  dollar,  and  the  product  is 

550  34375,  count  off  4  figures  for  decimals  of  a  dol- 

275  lar,  and  there  place  the  period,  and  you  have  one 

figure  (3)   at  the  left,  Avhich  is   3D.  43c.  .75. 


D3.4375  Ans.  And  12.5  pounds  at  27.5  cents  per  pound  would 
amount  to  the  same. 

8.  What  cost  30.5  pounds  of  nails  at  9c.  5m.  per  pound? 

Ans,  2D.  89c.  .75. 

9.  What  cost  42.25  pounds  of  leather  at  20c.  per  pound? 

Ans.  8D.  45c. 

10.  What  cost  21.5  pounds  of  flax  at  4c.  5m.  per  pound? 

Ans.  96c.  .75. 

11.  What  cost  32.5  yards  of  sheeting  at  14c.  5m.  per  yard  ? 

Ans.  AD.  71c.  .25. 

12.  What  cost  14.75  pounds  of  sugar  at  12c.  5m.  per  pound  I 

Ans.  ID.  84c.  3m.  .75. 

13.  What  cost  18.25  yards  of  calico  at  13c.  5m.  per  yard  ? 

Ans.  2D.  46c.  3m.  .75. 
14  What  cost  16  pounds  of  coffee  at  13c.  .75  per  pound  ? 

Ans.  2D.  20c. 

15.  What  cost  14  pounds  of  indigo  at  2D.  25c.  per  pound  ? 

Ans.  31D.  5Cc. 

16.  What  cost  4.25  yards  of  broadcloth  at  3D.  75c.  per  yard  ? 

Ans.  15D.  93c.  .75. 

17.  What  cost  6.5  yards  of  broadcloth  at  4D.  75c.  per  yard  ? 

Ans.  30D.  87c.  .5. 
J.S.  What  cost  3.5  bushels  of  wheat  at  ID.  50c.  per  bushel? 

Ans.  5D.  25c. 

19.  What  cost  18.22  bushels  of  oats  at  37.5c.  per  bushel? 

Ans.  6D.  83c.  .25. 

20.  What  cost  12.5  bushels  of  flaxseed  at  87.5c.  per  bushel? 

Ans.  lOD.  93c.  .75. 

21.  What  cost  4.25  bushels  of  cloverseed  at  4D.  25c.  per 
bushel?  Ans.  18D.  06c.  .25. 

22.  What  cost  742.5  pounds  of  pork  at  8.5c.  per  pound  ? 

Ans.  63D.  lie.  .25. 


FDLTIPLICATION    OF    DECIMALS. 


53 


23.  What  cost  434  pounds  of  ham  at  12.5c.  per  pound? 

Ans.  54D.  25c. 

24.  What  cost  75.5  pounds  of  lard  at  7.5c.  per  pound? 

Ans.  5D.  66c.  .25. 

25.  What  cost  630.5  pounds  of  cheese  at  6.25c.  per  pound? 

Ans.  39D.  40c.  .625. 

26.  What  cost  500  pounds  at  3m.  per  pound  ?    Ans.  ID.  50c. 

We  have  remarked,  that  prefixing  ciphers  to  a  decimal  dimin- 
ishes its  value  ten  times  for  every  cipher  prefixed.  Thus,  take 
i^"*^,  .05=y%%,  by  prefixing  one  cipher;  -005=^^^^  by  prefixing 
two  ciphers  ;  again,  the  value  of  a  decimal  figure  depends  on  its 
place f  or  distance  from  the  decimal  pom^,  whether  there  be  ciphers 
prefixed  or  not,  thus  :         .005  x  .005. 

.     5 
.000025 


Here  the  multiplicand  and  the  multiplier  are  the  same, 
that  is,  (five  thousandths) ;  the  ciphers  are  omitted  in  the  mul- 
tiplier, but  their  places  are  counted  in  the  product.  Multiply 
2.034  by  .014  =  .028476.  In  this  example  the  multiplicand  is 
a  w^hole  number  and  decimal,  vi^hile  the  multiplier  is  a  small 
decimal  (fourteen  thousandths)  ;  in  this  case  the  product  is  less 
than  the  multiplicand,  it  having  diminished  in  consequence  of 
the  small  value  of  the  multiplier,  it  being  many  times  less  than 
unity ;  it  follows  that  the  product  must  diminish  in  the  same 
proportion,  and  is  one  kind  of  subtraction.  When  a  decimal 
number  is  to  be  multiplied  by  10,  100,  1000,  &c.,  the  multipli- 
cation may  be  made  by  removing  the  decimal  point  as  many 
places  to  the  right  hand  as  there  are  ciphers  in  the  multiplier, 
and  if  there  be  not  so  many  figures  on  the  right  of  the  decimal 
point,  supply  the  deficiency  by  annexing  ciphers,  thus : — 


r  1^         1 

100 

r 

67.9 

. 

679. 

6.79  X< 

1000      y      = 

6790. 

10000      1 

67900. 

100000   J 

679000. 

*>.7, 

31.00467X10.03962. 

Ans. 

311.2751050254. 

M. 

596.04X0.00004. 

Afta. 

29. 

Multiply  341.45D.  by  .007. 

Ans. 

30. 

D.26.000: 

^75  X  .00007. 

Ans. 

.00182002625. 

31.  Multiply  three  hundred,  and  twenty-seven  hundredths 
by  31.  Ans.  9308.37. 

5* 


54  DIVISION"    OF    DECIMALS. 

DIVISION  OF  DECIMALS. 

The  operations  in  division  of  decimals  are  the  same  as  whole 
numbers,  with  the  exception  of  the  decimal  points.  We  have 
shown  in  multiplication  of  decimals,  that  one  decimal  multiplied 
by  another,  the  product  will  contain  as  many  places  of  decimals 
as  there  were  in  both  factors  ;  then  it  follows,  that  if  this  product 
be  divided  by  one  of  the  factors,  the  quotient  will  be  the  other 
factor  ;  therefore,  in  division,  the  dividend  must  contain  as  many 
places  as  the  divisor  and  quotient  together.  The  quotient  will 
contain  as  many  places  as  the  dividend^  less  those  of  the  divisor. 

RULE. 

1.  Write  down  the  dividend  and  divisor,  then  divide  in  the 
same  way  as  in  Simple  Division,  without  regard  to  the  decimal 
points.  2.  Whe.n  the  division  is  finished,  count  oiF  as  many 
ilgures  from  the  right  of  the  quotient,  as  the  decimal  figures  in 
the  dividend  exceed  those  in  the  divisor,  and  there  place  the 
decimal  point.  3.  If  the  decimal  figures  in  the  quotient  should 
be  less  than  is  required  above,  prefix  as  many  ciphers  to  the  left 
of  the  quotient  as  are  required,  and  there  place  the  decimal  point. 
4.  When  the  divisor  is  larger  than  the  dividend,  then  annex 
ciphers  until  it  exceeds  the  divisor  ;  then  place  the  decimal  point 
accordingly ;  when  there  is  a  remainder,  annex  ciphers  and 
continue  the  operation.  (See  examples,  &c.) 
1.  .1812)4.18000(23.0  Ans.  1812  In  this  example  we 
3624  23.0  X  have  a  whole  num- 
bej.  and  decimal  in 


5560  54360  the  dividend,  and  a 

5436  3624  decimal  for  a  divi- 

rem, \-     1240  sor ;    divide  as   in 

Rem.  1240  Long  Division,  and 

Proof,     4.1 8000  you  have  three  quo- 

tient figures,  a  part  of  which  must  be  a  whole  number  ;  therefore, 
by  the  rule,  you  will  observe  you  have  5  places  of  decimals  in 
the  dividend  and  4  in  the  divisor,  difference  1  ;  therefore  count 
ofif  from  the  right  one  place,  and  there  make  the  point. 
2.  .957)7.25406(7.58  In  this  example  we  have  5  decimal 
6699  places  in  the  dividend  and  3  in  the 

divisor,    difference    2 ;     therefore, 

5550  count  off*  2  places  from  the  right 

4785  in  the  quotient,  and  there  make  the 

point. 

7656 
7656 


I 


DIVISION  OF    DECIMALS.                                          55 
D.    d. 

3.  D.  304  81)186513.239(611.9     In  this  example  the  denom 

182886  inations  are  dollars,  deci 

mals  of  the  Doll,  unit , 

36272  we  find  3  decimal  piaces 

30481  in  the  dividend  and  2  in 

the  divisor,  difference  1  ; 

57913  therefore  count  one  place 

30481  from    the    right    in   the 

quotient,  and  there  make 

274329  the  point,  and  you  have 

274329  611D.   9=:/^D.  =  9d.= 


90c. 


4.  Divide  600D.  equally  among  25  men.  Ans.  24D. 

5.  Two  men  received  D235.25  ;   what  is  the  share  of  each? 

Ans.  117D.  62c.  .5. 

6.  A.  bought  1584  yards  of  sheeting  for  250D.  ;   how  much 
was  it  per  yard  ?  Ans.  15c.  7m. 

7.  A  man  was  paid  for  his  labor  of  26  days,  16D.  75c. ;  how 
much  did  he  earn  daily  ?  Ans.  64c.  4m. 

8.  Seven  men  have  the  sum  of  2764D.  25c. ;   how  much  did 
each  receive?  Ans.  394D.  89c.  3m. 

9.  A  farmer  purchased  75  sheep  for  255D.  ;   how  much  did 
he  pay  per  head?  Ans.  3D.  40c. 

10.  B.  gave  150D.  for  60  bushels  of  wheat;  how  much  was 
it  per  bushel  ?  Ans.  2D.  50c. 

11.  If  I  paid  400D.  for  64  yards  of  broadcloth,  how  much 
was  it  per  yard  ?  Ans.  6D.  25c. 

12.  How  many  times  are  12.5c.  contained  in  125D.  ? 

Ans.  1000. 

13.  How  many  half  dimes  in  75D.  ?  Ans.  1500. 

14.  Paid  20D.  for  130  pounds  of  sugar ;  how  much  per  pound  ? 

/  Ans.  15c.  3m. 

15.  Paid  25D.  for  400  pounds  of  flax ;  how  much  was  it  per 
pound?  Ans.  6c.  .25 

16.  Gave  75D.  for  200  bushels  of  oats  ;  what  is  the  cost  per 
bushel  ?  Ans.  37Jc. 

17.  A  farmer  raised  480  bushels  of  grain  from  30  acres; 
how  many  bushels  per  acre  ?  Ans.  16. 

18.  C.  paid  750D.  for  25  i.:ows  ;  what  cost  1  cow  ?  Ans.  SOD. 

19.  If  a  man  spend  250D.  in  a  year,  how  much  is  that  daily  ? 

Ans,  68.5c.  (nearly). 

20.  If  wool  is  worth  37.5c.  per  pound,  how  much  can  you 
buy  for  500D.  ?  Ans.  1333  pounds,  .3=,^^+ 


56  DIVISION    OF    DECIMALS. 

When  there  are  more  decimal  places  in  the  divisor  than  in 
the  dividend,  annex  as  many  ciphers  to  the  dividend  as  are 
necessary  to  make  its  decimal  places  equal  to  those  of  the  divi' 
sor ;  all  the  figures  of  the  quotient  will  then  be  whole  numbers, 
but  if  there  be  a  remainder  and  the  operation  continued,  then 
the  quotient  figure  will  be  decimal. 

Example  1.— Divide  8794.8  by  6.98  : 
6.98)8794.80(1260  Ans. 

698  In  this  example  there  is  one  decimal 

place  in  the  dividend,  and  two  in  the 

1814  divisor;  therefore  you  will  annex  a  ci- 

1396  pher  to  the  dividend,  and  this  will  make 

the  decimal  places  equal,  and  the  quo 

4188  tient  a  whole  number. 

4188 


Example  2. — Divide  4.64  by  2.22  : 
2.22)4.64(2.09 

4  44  In  this  example  you  annex  a  ciphei 

to  the  remainder  ;  the  next  quotient  ^g 

2000  ure  will  be  a  decimal. 

1998 


Rem.    2 
When  circulating  decimals  occur,  it  will  be  sufficient  to  con- 
tinue the  division  to  three  or  four  places,  unless  great  accuracy 
is  required,  then  it  may  be  continued  farther  ;  but  it  is  indef- 
inite, and  will  never  terminate. 

Examples  9)10.00(1.11+  .06).20(3.33+  6)10.00(1.66  + 
When  a  decimal  number  is  to  be  divided  by  10,  100,  1000, 
<fec.,  the  division  is  made  by  removing  the  decimal  point  as 
many  places  to  the  left  as  there  are  ciphers  in  the  divisor ;  and 
if  there  be  not  so  many  figures  on  the  left  of  the  decimal  point, 
the  deficiency  must  be  supplied  by  prefixing  ciphers. 

10        1        r4.274  101   bo  r  ■].2f765.4 

7654  I J 


U 


fe  .    1  100       I        I     .4274  100  ,  ;S   ]  ^c^^A  I  S  J  76.54 


101 

bO 

100 
1000 

> 

10000  J 

n3 

;2"^'{  1000     f  -  ]     .04274         1000  f  T  1             f  "■§  1  7.654 

[  10000  J        {    .004274     lOOOOJ  ^  {           J   §.[-7654 

21.  Divide  33.66431  by  1.01.  Ans,  33.331. 

22.  Divide  .01001  by  .01. 

23.  Divide  2194.02194  by  .100001.  Ans.  21940. 

24.  Divide  .1  by  .0001.  Ans,  1000 

25.  Divide  37.4  by  4.5.  Ans,  8.311 1-f 


APPLICATION    OF    DECIMALS.  '     57 

26.  Dmde  94.0369  by  81.032.  Ans,  1.160-f 

27.  If  2D.  25c.  will  board  one  man  a  week,  how  man)'*  weeks 
can  he  be  boarded  for  lOOlD.  25c.?  Ans.  445. 

28.  If  3.355  bushels  of  corn  will  fill  one  barrel,  how  many 
barrels  will  352.275  bushels  fill?  Ans,  105. 

,29.  1561.275~-24.3r=6425  ^n;?.       30.   .264^2=. 132  An;?. 


APPLICATION  OF  DECIMALS. 

1.  Add  7482D,  97c.  6m.  .50;  4642D.  34c.  9m.  .50  ;  3765D. 
37c.  5m.  .25 ;  2320D.  63c.  7m.  .37,  together. 

Ans.  18211D.  33c.  8m.  .62. 

2.  What  cost  7.25  yards  of  broadcloth  at  6D.  50c.  per  yard  ? 

Ans,  47D.  12c.  .5. 

3.  What  cost  7.25  bushels  of  wheat  at  ID.  37c.  .5,  per 
bushel  ?  Ans.  9P,  96c.  8m.  .75. 

4.  What  cost  9.75  bushels  of  wheat  at  ID.  62c.  .5,  per  bush- 
el? Ans.  15D.  84c.  3m.  .75. 

5.  What  cost  15.35  bushels  of  com,  at  87c.  .5,  per  bushel  ? 

Ans.  13D.  43c.  Im.  .25. 

6.  A  man  purchased  1500  pounds  of  cheese,  for  which  he 
paid  6c.  .25  per  pourwl ;  it  cost  him  7D.  50c.  to  convey  it  to 
market ;  he  then  disposed  of  it  for  8c.  .5  per  pound ;  did  he 
gain  or  lose,  and  how  much  ^  Ans,  gained  26D.  25c. 

7.  If  you  buy  2000  bushels  of  wheat  for  2500D.  and  sell  it 
for  ID.  45c.  per  bushel,  how  much  will  you  gain  in  all  ?  and 
how  much  on  a  bushel  ? 

Ans.  gain  in  all,  400D.,  which  is  20c.  per  bushel, 

8.  James  Wilson,  Dr, 

To  4.5  pounds  of  candles,  at  12c.  .5  per  pound. 
To  9  pounds  of  soap,  at  8c.  .25  per  pound. 
To  12.25  yards  of  calico,  at  16c.  .75  per  yard. 
To  11.5  pounds  of  butter,  at  12c.  .5  per  pound. 
Required  the  amount  of  the  above  bill.  A71S.  4D.  79c.    4. 

9.  John  Thompson,  Dr. 

To  6.5  bushels  of  oats,  at  37.5c.  per  bushel. 
To  8.75  pounds  of  feathers,  at  43c,  per  pound. 
To  12.5  pounds  of  coffee,  at  16c.  per  pound. 
To  1  side  of  upper  leather,  at  3D.  25c. 
To  1  set  of  knives  and  forks,  at  2D.  25c. 
There  has  been  paid  4D.  75c.  ;  required  the  balance  due. 

Ans.  8D.  95c 


68  RETIEW    OF    DECIMALS. 

10.  A.  had  an  order  on  a  merchant  for  87D.  50c.  to  be  paid 
in  goods  ;  after  purchasing  the  following  articles,  how  will  the 
balance  stand,  viz.  :  5  sets  of  cups  and  saucers  at  75c.  per  set; 
38  yards  of  sheeting  at  14.5c. ;  8  gallons  of  wine  at  ID.  25c. 
per  gallon  ;  4.25  yards  of  broadcloth  at  5D.  25c.  per  yard  ;  4.5 
pounds  of  tea  at  ID.  25c.  per  pound ;  9.5  bushels  salt  at  62.5c« 
per  bushel;  18  pounds  of  nails  at  12.5c.  per  pound  ;  4  barrels 
of  flour  at  6D.  25c.  per  barrel  1 

Ans.  balance  due  on  order  7D.  11.5. 

11.  Divide  764D.  76c.  among  4  men  and  a  boy. 

12.  Multiply  500D.  by  500  cents. 

13.  What  will  225  bushels  of  wheat  cost  ot  ID.  12.5c.  per 
bushel?  Ans.  253D.  12.5c. 

14.  What  will  47.25  acres  of  land  cost  at  54D.  per  acre  ? 

Ans  2551D.  50c. 

15.  If    1  gallon  of  wine  is  w^orth  ID.  87.5c.,  wbat  are  20  , 
gallons  worth  ?  Ans.  37D.  50c. 

16.  If  the  income  of  an  estate  is  3650D.  40c.  per  annum, 
how  much  for  one  day  ?  Ans.  lOD.  00c.  Im. 

17.  How  much  would  it  be  for  1  week  ?  for  1  month  ?  &;c. 

18.  If  383  yards  of  cloth  cost  50D.36.5c.  how  much  for  one 
yard  1  Ans.  OD.  13.15c. 

19.  If  you  can  earn  600D.  in  a  year,  how  much  is  it  a  week  ? 

Ans.  1  ID.  54c 

20.  Divide  10489  by  .9846=10653.0+  ;  8.47-f-.7=12.1. 

In  the  last  two  examples,  where  the  dividend  is  a  whole  num- 
ber and  the  divisor  is  a  decimal,  the  quotient  must  consist  of 
one  figure  more  than  the  divisor,  which  will  be  a  whole  number, 
and  the  next  figure  will  be  a  decimal. 

REVIEW  OF  DECIMALS,  NO.  I. 

What  do  you  understand  by  decimal  ?  What  is  a  fraction  ? 
When  ciphers  are  annexed  to  a  decimal,  do  they  increase  the 
value  of  the  decimal  ?  What  is  the  value  when  prefixed  ?  Are 
there  more  than  one  kind  of  decimals  ?  What  is  a  circulating 
decimal  *?  Is  the  currency  of  the  United  States  decimal  ?  Why  ? 
What  do  the  laws  of  the  country  require  in  relation  to  accounts, 
book-keeping,  &c.  ?  When  the  unit  1  is  divided  into  10  equal 
parts,  what  is  each  part  called  ?  When  in  money  of  the  United 
States,  how  are  they  called  "?  Is  it  necessary  to  set  down  the 
denominator  of  a  decimal?  If  you  should,  would  it  make  any 
difference  in  the  value  ?  What  is  the  place  next  to  the  decimal 
point  called  ?  The  second  ?  The  third  ?  How  do  you  nu- 
merate decimals  ?     Does  the  value  of  a  figure  depend  upon  the 


b 


REVIEW    OF    DECIMALS.  69 


distance  from  the  decimal  point,  or  the  place  which  it  occupies  ? 
How  does  the  value  change  from  the  left  toward  the  right  ? 
What  part  of  a  dollar  is  a  dime  ?  A  cent  ?  A  mill  ?  Does  the 
annexing  of  ciphers  to  a  decimal,  alter  its  value  ?  When  a  ci- 
pher is  prefixed  to  a  number?  When  prefixed,  does  it  increase 
the  numerator  or  denominator  ?  Will  it  produce  any  effect  on 
the  value  of  a  fraction  ?  What  will  .8  become  by  prefixing  a 
cipher  1  How  do  you  write  down  numbers  in  addition  ?  How 
are  you  to  place  the  decimal  points  ?  How  will  you  then  pro- 
ceed ?  Where  will  you  place  the  decimal  point  in  the  sum  ^ 
Repeat  the  rule  ?  Give  an  example  in  addition  on  your  slate. 
What  is  the  rule  for  Subtraction  of  Decimals  ?  How  do  you 
write  down  the  numbers  1  How  will  you  then  proceed  ?  Where 
will  you  place  the  decimal  point  in  the  remainder  ?  What  is 
the  rule  for  Multiplication  of  Decimals  ?  After  multiplying, 
how  many  decimal  places  will  you  count  off  in  the  product  ? 
When  there  are  not  so  many  in  the  product,  what  will  you  do  ? 
How  do  you  multiply  a  decimal  by  10,  1000,  &c.  1  What  more 
can  you  say  of  Multiplication  of  Decimals  ?  Give  a  few  ex- 
amples on  your  slate.  What  is  the  rule  for  Division  of  Deci- 
mals ?  How  does  the  number  of  decimal  places  in  the  dividend 
compare  with  those  in  the  divisor  and  quotient  ?  How  do  you 
determine  the  number  of  decimal  places  in  the  quotient  ?  Sup- 
pose there  are  two  places  in  the  divisor,  and  three  in  the  divi- 
dend, how  many  would  you  count  off  in  the  quotient  ?  How  do 
you  divide  a  decimal  by  10,  100,  &c.  1  When  there  are  more 
decimal  places  in  the  divisor  than  is  in  the  dividend,  what  will 
you  do  ?  In  this  case  what  will  the  figures  in  the  quotient  he  ? 
How  can  you  continue  division  after  you  have  brought  down  all 
the  figures  in  the  dividend  ?  Give  a  few  examples  of  your 
knowledge  of  decimals  on  the  slate,  with  an  explanation  of  the 
same.  Divide  .2142  by  3.2  =  .066+.  Divide  2.00385  by  931 
=  .0021523.  Can  there  be  a  decimal  figure  in  the  quotient, 
unless  you  have  a  decimal  in  the  dividend,  or  by  annexing  a 
cipher,  which  is  the  same  ?  When  you  have  brought  down  a 
decimal  figure  to  the  remainder,  what  will  the  next  quotient  fig- 
ure be  ?  When  the  price  of  one  article  is  given,  and  you  wish 
to  know  the  value  of  100,  1000,  &c.,  how  will  you  place  the 
decimal  point  without  midtiplying? 

Ans.  Thus,  1  yard,  37.5c.  ;  10  yards,  D.3.75  ;  1000  yards, 
D.375.00,  (fee. 

The  denominator  to  a  decimal  fraction,  although  not  express- 
ed, is  always  understood. 


60  TABLES    OF    WEIGHTS    AND    MEASURES. 

TABLES  OF  WEIGHTS  AND  MEASURES. 

Let  the  pupil  commit  all  the  tables  to  memory. 

AVOIRDUPOIS  WEIGHT. 

By  this  weight  are  weighed  things  of  a  coarse  or  drossy  na- 
ture, and  all  metals,  except  silver  and  gold. 

Denominations.  Marked, 

16  drams  (dr.)         equal,  or  make  1  ounce,                    oz. 

16  ounces                                    "  1  pound,                     lb. 

28  pounds                                    "  1  qr.,    (gross    weight.) 

4  quarters,  or  112  lb.               "  1  hund.cwt.(grosscwt.) 

100  pounds  (in  general  use)      "  1  hund.  C.         (lb.  wt.) 

20  hund.  wt.  (2240  lb.)            "  1  ton                           T. 

TROY  WEIGHT. 

Gold,  silver,  jewels,  and  liquors,  are  weighed  by  this  weight. 

Denominations,  Marked. 

24  grains  (gr.)  make  1  pennyweight,  pwt.  or  dwt. 

20  pennyweights  "  1  ounce,  oz. 

12  ounces  "  1  pound,  lb. 

Note. — 1  lb.  troy =5760  grains,  and  1  lb.  avoirdupois —  7000 
grains,  175  oz.  troy  =:  192  avoirdupois  oz.,  175  lb.  troy=144 
avoirdupois  lb. 

APOTHECARIES'  WEIGHT. 

By  this  weight  apothecaries  mix  their  medicines,  but  buy  and 
sell  by  avoirdupois  weight. 

Denominations. 

20  grains  (gr.)  make  -  1  scruple. 

3  scruples  "  1  dram. 

8  drams  "  1  ounce. 

12  ounces  "  1  pound. 

Note. — The  lb.  and  oz.  apothecaries'  weight,  and  the  lb.  and 
z.  troy,  are  the  same,  only  differently  divided  and  subdivided. 

CLOTH  MEASURE. 

By  this  measure  cloth,  tape,  ribands,  calico,  &;c.,  are  measured. 
Denominations.  Marked, 

2.25  (2^)  inches,  (in.)  make  1  nail.  na, 

4  nails,  or  9  in.  "1  quarter  of  a  yard>         qr 


TABLES?    OF    WEIGHTS    AND    MEASURES. 


ai 


n 


Denaminations, 

Marked. 

10  nails,  or  22^  in. 

make 

1  ell  Hamburgh, 

E.  H. 

3  quarters,  or  27  in. 

(( 

1  ell  Flemish, 

E.  F. 

27.2  inches 

*• 

1  ell  Scotch, 

E.  S. 

33  inches 

<( 

1  var  Spanish, 

V.  S. 

4  quarters,  or  36  in. 

u 

1  yard. 

yd. 

5  quarters,  or  45  in. 

C( 

1  ell  English, 

E.  E. 

^   6  quarters,  or  54  in. 

{< 

1  ell  French, 

E.  F. 

The  yard  is  used  in  measuring  all  kinds  of  piece  goods  in  the 
nited  States. 

LONG  MEASURE. 
This  measure  is  applied  to  distance,  or  length. 
Denominations. 


Marked. 


make  1  inch, 

in. 

(( 

1  foot. 

ft. 

ti 

1  yard. 

yd. 

u 

1  rod,po.. 

or  perch,  R.  P. 

(( 

1  furlong. 

fur. 

0  yds 

1  mile, 

M. 

3    barleycorns  (be.) 
12    inches 

3    feet,  or  36  in. 

H  (5.5)  yards,  or  16i  ft. 
40    poles,  or  220  yards. 


69^  (69.5)  statute,  or  )    ^  ^^^  ^  ^ 

60    geographic  J  '  ^      '  ^ 

360    degrees  "     1  circle,  cir. 

A  circle  is  the  circumference  of  the  earth. 
6  points  =1  line,  12  lines  1  inch,  applied  to  the  measure  of 
pendulums. 

Note. — A  hand  is  4  inches,  used  in  measuring  the  height  of 
horses  ;  a  fathom  is  6  feet,  used  in  measuring  the  depth  of  wa- 
ter (fm.)  ;  a  league  is  3  miles,  used  in  measuring  distances  at 
sea  (L.). 

LAND  OR  SQUARE  MEASURE. 

This  measure  is  applied  to  land,  and  has  respect  to  length 
and  breadth,  but  not  to  depth. 
Denominations.  Marked. 


144  square  inches,  (in.) 
9  square  feet  (1296  in.) 

30.25  sq.  yds.  or  272.25  sq.  ft. 
40sq.  P.  (1210sq.  yds.) 

4  roods,  or  160  P. 
640  acres  or  1  section  of  land, 


1  foot 
square 


rnUe 

.1 1 1 11 


9  ft.  or  sq.  yd. 


make  1  square  foot,     ft. 
"      1  sq.  yard,  sq.  yd. 

"      1  sq.  perch,  sq.  P. 
1  rood,  R. 

1  acre,  A. 

1  square  mile,   M. 


6a 


TABLES    OF    WEIGHTS    AND    MBASXTRES. 


CHAIN. 


Denominations. 


Marked 


7.92  inches         -                 -  equal  1  link,                       li. 

25  links                 -                 .  «     1  rod,pole,orpeTch,  P 

4  rods,  or  6Q  ft.  Gunter*s  ch.,  or  100  li.      1  chain,  ch. 

1  square  chain                      -  **  16  square  poles. 

80  chains,  or  320  rods           -  "1  mile,                      m. 

10  square  chains  -                 -  "1  acre,                       A. 

The  American  or  English  mile  is  5280  feet. 

French  mile           -                 -  5328     " 

Italian  mile            -                  -  5566     " 

German  mile         -                 -  26400     " 

Dutch,  Spanish,  or  Polish  mile  21120     " 

Scotch  mile           -                 -  7920     " 

Indian  mile            -                 -  15840     " 

Irish  mile              -                 -  6720     " 


LIQUID  MEASURE. 

This  measure  is  used  for  beer,  wine,  cider,  &c.     The  gallon 
contains  231  cubic  inches,  or  .13368  feet. 

Denominations,  Marked, 


4    gills  (gi.) 

- 

equal  1  pint. 

pt. 

2    pints 

- 

"     1  quart. 

qt 

4    quarts 

- 

"     1  gallon. 

gal. 

16    gallons 

. 

"     1  half-barrel, 

hf.  bar 

31^  (31.5)  gallons- 

- 

"     1  barrel. 

bar 

42    gallons 

• 

"     1  tierce, 

tier 

63    gallons 

- 

"     1  hogshead, 

hhd 

84    gallons 

- 

"     1  puncheon, 

punch 

2    hogsheads 

- 

"     1  pipe  or  butt, 

p.  or  b. 

2    pipes  or  butts  - 

- 

"     1  tun, 

T. 

DRY  MEASURE. 

This  measure  is  used  for  all  dry  goods,  or  such  as  are  sold 
by  the  bushel. 

Denominations,  Marked 

2  pints  (pt.)  -  -    equal  1  quart,  qt. 

8  quarts  -  -       "     1  peck,  pk. 

4  pecks  =  32  qts.=i64  pts.     -       "     1  bushel,  bush. 

i96  pounds  flour,  200  pounds  pork,  '*     1  barrel,  bar. 

A  bushel  is  18  J  inches  in  diameter,  and  8  inches  in  depth. 


TABLES    OF    WEiOHTS    AND    MEASURES.  63 

MOTION  OR  CIRCLE  MEASURE. 

This  measure  is  used  by  astronomers,  navigators,  &c. 

Denominations.  Marked, 

60  seconds  ('')  equal    1  minute,  ' 

60  minutes  -         -  **       1  degree,  ° 

30  degrees        -         -         -  "1  sign,  sig. 

12  sitjns,  360  degrees        -  "1  revolution  of  a ) 

°  1       *        •    1    c    rev. 

planet  or  circle ) 

Sound  moves  at  the  rate  of  1142  feet  per  second,  equal  to 

12  miles,  3  furlongs,  32  poles,  4  yards,  per  minute.  Light 
passes  from  the  sun  to  the  earth,  95  millions  of  miles,  in  8.2 
minutes,  or  11585365  miles  per  minute. 

SOLID  OR  CUBIC  MEASURE. 

By  this  measure  is  ascertained  the  solid  content  of  all  things 
in  which  length,  breadth,  and  thickness,  are  considered ;  such 
as  stone,  timber,  wood,  bark,  grain,  coal,  &c,  A  cube  is  a  solid 
of  six  eq  lal  sides. 

Denominations.  Marked. 

1728  cubic  inches  (c.  in.)  equal    1  cubic  foot,        c.  ft. 

27  cubic  feet,  or  46656  in.  "      1  cubic  yard,       c.  y. 

40  feet  of  round  timber,  or  j 

50  feet  of  hewn  timber, 
2150.4252  c.  in.,  or  1.24446  c.  ft.        "       1  bushel,  of  strick 

measure,        c.bu. 
2553.6299  c.in.,or  1.47779+ eft.     "      1  bush,  of  heaped 

measure. 
128  cubical  feet,  or  221 184  c.  in.,] 

or  8  feet  long,  4  feet  wide,  and  ^    **       1  cord  of  wood,    C. 

4  ft.  high:  8x4=32  X  4=128,  j 
A  cord  foot  is  one  foot  in  length  of  the  pile  which  makes  a 

cord,  and  is  equal  to  16  solid  feet,  or  -I-  of  a  cord. 
]  6^  solid  feet=  1  solid  perch. 

TIME. 
Denominations. 
60  seconds,  sec. 
60  minutes,    -         -        - 
24  hours,        -         -         - 

7  days,  -         - 

4  weeks       .         -         - 

13  months,  1  day,  6  hours  (lunar),  or 
12  mo.  of  30.4375  days  each,  or 
365.25,  365J  days,  or  52  weeks, 


1  ton,  T. 


Marked 

equal  1  minute, 

m. 

"     1  hour, 

hr. 

"     1  day. 

da.  D. 

"     1  week, 

w. 

"     1  mo.  lunar, 

mo. 

>"     1  Julian  year. 

64 


TABLE    OF    WEIGHTS    AND    MEASURES 


365  days,  5  hours,  48  minutes,  57  seconds,  or  365.2423+ 
=  1  solar  year,  100  years  =1  century.  Every  4th  or  leap-year 
has  366  days. 

The  year  is  also  divided  into  12  calendar  months,  as  follows : 


Istmonth,  January,  has  31  days. 

7th  month, 

July,  has  31  days 

2d      "       February, 

28     " 

8th     " 

August,     31     ** 

3d      "       March, 

31     " 

9th     « 

Sept.,        30     " 

4th     "       April, 

30     " 

10th  " 

Oct.,          31     " 

5th     "       May, 

31     " 

11th  " 

Nov.,        30     " 

6th     "       June, 

30     " 

12th  *' 

Dec,         31     " 

The  diurnal  motion  of  the  earth  on  its  axis  is  17^  miles  per 
minute  at  the  equator=1035  miles  per  hour.     The  earth  moves 
in  its  orbit  68249  miles  per  hour=  1637965.6  per  day. 
"  Thirty  days  hath  September, 
April,  June,  and  November  ; 
February  twenty-eight  alone  ; 
All  the  rest  have  thirty-one." 
N.  B. — In  bissextile,  or  leap-year,  February  has  29  days. 
Note. — To  know  when  it  is  leap-year,  divide  the  years  over 
even  centuries  by  4  ;  if  no  remainder,  it  is  leap-year  ;  but  if  any 
remain,  it  is  so  many  years  after  leap-year.     (See  the  note  at 
the  close  of  the  tables.) 


Denominations. 
24  sheets,  or  23.6  (sh.) 
20  quires 
2  reams 


PAPER. 


make 


1  quire, 
1  ream, 
1  bundle, 


Marked. 
qr. 
re. 
bun. 


PARTICULARS, 

Denominations. 

12  single  things 

12  dozen 

12  gross  (144  dozen)     - 


Marked. 
make  1  dozen,  doz. 

**      1  gross,  gro. 

1  great  gross,  g.  gro. 


20  single  things  (112  lbs.=:l  quintal  fish)  1  score,  sco. 

5  scores  (2  are  1  pair,  or  couple)     "      1  hundred,         hund. 

BOOKS. 

Denominations.  Marked 

36mo.  is  when  1  sheet  makes  36  leaves,  or  72  pages,      pp. 


24mo.                  1 

it 

24 

48       " 

18mo.                  1 

u 

18 

36       " 

12mo.                  1 

« 

12 

24       " 

Octavo,  or  8vo.  1 

i( 

8 

16       " 

Quarto,  or  4to.  1 

tt 

4 

.   8       " 

Folio,  or  fol.      1 

ti 

2 

(( 

4       " 

TABLES    OF    WEIGHTS    AND    MEASURES.  65 

Note.  Extracts :  The  civil  solar  year  of  365  days  being 
Bhort  of  the  true  year  by  5h.  48m.  48sec.,  occasioned  the  be- 
ginning of  the  year  to  run  forward  through  the  seasons  nearly  1 
day  in  4  years.  On  this  account  Julius  CcBsar  ordained  that 
one  day  should  be  added  to  February  every  4th  year,  by  causing 
the  24th  day  to  be  reckoned  twice ;  and  because  this  24th  day 
was  the  sixth  (sextilis)  before  the  Kalends  of  March,  there  were 
in  this  year  two  of  these  sextiles^  which  gave  the  name  of  Bis^ 
sextile  to  this  year,  which  being  thus  corrected,  was  thence 
called  the  Julian  year. 

Pope  Gregory  the  13th  made  a  reformation  of  the  calendar. 
The  Julian  calendar,  or  old  style,  had,  before  that  time,  been  in 
general  use  all  over  Europe.  The  year,  according  to  the  Julian 
calendar,  consists  of  365  days  and  6  hours  ;  which  6  hours  being 
one  fourth  part  of  a  day,  the  common  years  consisted  of  365  days, 
and  every  fourth  year,  one  day  was  added  to  the  month  of  Feb- 
ruary, which  made  each  of  those  years  366  days,  which  are  usu- 
ally called  leap-years.  This  computation,  though  near  the  truth, 
is  more  than  the  solar  year  by  eleven  minutes  which,  in  one  hun- 
dred and  thirty-one  years  will  amount  to  a  whole  day.  By  this 
calculation,  the  vernal  equinox  was  anticipated  ten  days  from 
the  time  of  the  general  council  of  Nice,  held  in  the  year  325  of 
,lhe  Christian  era  to  the  time  of  Pope  Gregory,  who  therefore 
caused  ten  days  to  be  taken  out  of  the  month  of  October  iu  1582, 
to  make  the  equinox  fall  on  the  21st  of  March,  as  it  did  at  the 
time  ot  that  council.  And  to  prevent  the  like  variation  for  the 
future,  he  ordered  that  three  days  should  be  abated  in  every  four 
hundred  years,  by  reducing  the  leap-year  at  the  close  of  each 
century,  for  three  successive  centuries,  to  common  years,  and 
retaining  the  leap-year  at  the  close  of  each  fourth  century  only. 
This  was  at  that  time  esteemed  as  exactly  conformable  to  the 
true  solar  year ;  but  Dr.  Halley  makes  the  solar  year  to  be  365 
days,  five  hours,  forty-eight  minutes,  fifty-four  seconds,  forty-one 
thirds,  twenty-seven  fourths,  and  thirty-one  fifths  ;  according  to 
which,  in  400  years,  the  Julian  year  of  365  days  6  hours,  will 
exceed  the  solar  year  by  three  days,  one  hour,  and  fifty-five 
minutes,  which  is  near  two  hours,  so  that  in  50  centuries  it  will 
amount  to  a  day,  that  is,  the  Ih.  55m.  in  400  years,  would  in  50 
centuries=23h.  57m.  30sec. ;  +3  daysr=3d.  23h.  57m.  30sec. ; 
which  would  be  2>Jm.  less  than  4  days,  and  would  be  the  true 
difference  in  5,000  years.  Though  the  Gregorian  calendar,  or 
new  style,  had  long  been  used  throughout  the  greatest  part  of 
Europe,  it  did  not  take  place  in  Great  Britain  and  America  till 
^lie  first  of  January,  1752  ;  and  in  September  following,  the 


66  SIMPLE    REDUCTION. 

eleven  days  were  adjusted,  by  calling  the  third  day  of  that  month 
ihe  fourteenth,  and  continuing  the  rest  in  their  order. 

A  just  and  equal  measure  of  the  year  is  called  the  periodical 
year,  as  being  the  time  of  the  earth's  period  about  the  sun,  in 
departing  from  any  fixed  point  in  the  heavens,  and  returning  to 
the  same  again.  The  Zodiac  is  a  great  circle  of  the  sphere, 
containing  the  twelve  signs,  through  which  the  sun  passes. 


SIMPLE  REDUCTION. 

There  are  two  kinds  of  Reduction,  termed  simple  and  com- 
pound, by  which  we  are  taught  how  to  change  a  sum  or  quantity 
of  one  denomination  to  another,  whether  it  be  greater  or  less,  still 
retaining  the  same  value  ;  when  the  sum  consists  of  only  one  de- 
nomination, and  that  is  to  be  changed,  or  reduced  to  another,  it  is 
called  Simple  Reduction.  The  operations  are  all  performed  either 
by  multiplication  or  division  ;  when  by  multiplication,  it  is  called 
reduction  descending;  when  by  division,  reduction  ascending; 
for  instance,  to  reduce  pounds  (avoir.)  to  drams,  multiply  by  16, 
which  will  reduce  it  to  ounces,  then  by  16  again,  and  it  will  be 
in  drams  ;  then  take  those  drams  and  divide  by  16,  and  it  will  be 
ounces  ;  then  divide  again  by  16,  and  it  will  be  pounds,  because 
16  ounces  make  1  pound,  and  16  drams  1  ounce,  &c.  By  the 
above  process,  sums  in  reduction  will  reciprocally  prove  each 
other ;  and  it  would  be  well  for  the  teacher  to  require  it  of  the  pupil. 

RULE. 

1 .  Multiply  the  sum  or  quantity  by  that  number  of  the  next 
lower  denomination  which  it  requires  to  make  one  of  its  own. 

2.  If  there  be  one  or  more  denominations  between  the  de- 
nomination of  the  given  sum  or  quantity,  and  that  to  which  it  is 
to  be  reduced,  first  reduce  it  to  the  next  lower  than  its  own,  and 
then  to  the  next  lower,  &c. 

3.  When  low  denominations  are  to  be  brought  to  higher  de- 
nominations, as  for  example,  drams  to  pounds,  cents  to  dollars, 
inches  to  miles,  &;c.,  divide  by  as  many  of  the  lower  as  make 
one  of  the  higher,  and  set  down  what  remains  (if  any)  at  the 
right ;  so  proceed  till  you  have  brought  it  into  that  denomination 
which  your  question  requires.     See  example. 


I 

SIMPLE    REDUCTION.  6T 

Example.     Bring  2cwt.  to  drams. 

2  cwt.  Explanation.  First  multiply  tho  2cwt. 

4xqr.  =  l  cwt.      by  4,  and  this  will  reduce  it  to  quar- 
-  ters,  because   4  quarters  =  1  cwt , 

8  qr.=2  cwt.         then  multiply  the  quarters  by  28,  and 
28xlb.=l  qr.         this  will  reduce  it  to  pounds,  because 

28  lb.=:l  qr. ;  then  multiply  by  16 

224  lb.  =  8  qr.  which    will    reduce    it    to    ounces 

16xoz.=l  lb.         because   16  ounces  =  1   lb.;    then 

S"     multiply  the   ounces  by   16,  which 

ft      1344  will  reduce  it  to  drams:  because  16 

5^      224  drams  =  1    ounce,    &c.     Then    to 

Q      prove  it,  take  the  sum  of  the  drams, 

3584  oz.=224  lb.        and  divide  by  the  same   denomina- 
16xdrams=l  oz.   tions  by  which  you  multiplied,  and 

this  will  bring  back  the  2  cwt. 

16)57344  drams =3584  oz. 


^JO 


16)3584  ounces. 


.1^  28)224  pounds. 

Si         ~~ 

V3         4)8  quarters. 

2  cwt.  proof. 
Bring  1  year  to  seconds.  Ans»  31536000 

Thus  ;  1  year =365  days. 
4- 


1460 
6 


4x6=24  hours =1  day. 


8760  hours. 

60  minutes=l  hour. 


525600  minutes. 

60  seconds=l  minute. 


6,0)3153600,0  seconds.  Ans, 
6,0)52560,0  minutes. 


6)8760  hours. 

4)1460(365  days=l  year,  proof. 


i 

68  SIMPLE    REDUCTION. 

Avoirdupois  Weight. 

1.  In  120  pounds,  how  many  ounces  ^  Ans.  1920 

2.  Bring  1  cwt.  to  ounces.  1792 

3.  Bring  10  cwt.  to  ounces.  17920. 

4.  Bring  1284  ounces  to  pounds.  80  lbs.  4  oz. 

5.  Bring  1642  quarters  to  cwts. 

6.  Bring  184  drams  to  ounces. 

7.  Bring  1674  drams  to  pounds 

8.  Bring  1  ton  to  drams. 

Troy  Weight, 

9.  Reduce  15  lbs.  to  ounces.  Ans,  180. 

10.  Bring  20  lbs.  to  grains.  115200. 

11.  Bring  35  ounces  to  grains.  16800. 

12.  Bring  1476  ounces  to  pounds.  123. 

13.  Bring  34  lbs.  to  ounces.  408 

14.  Bring  97  lbs.  to  grains. 

15.  Bring  1796  grains  to  ounces. 

16.  Bring  1100  ounces  to  pounds. 

Apothecaries^  Weight, 

17.  Bring  72  ounces  to  drams.  Ans,  576. 

18.  Bring  10  pounds  to  grains.  57600. 

19.  Bring  17  pounds  to  ounces.  204. 

20.  Bring  57  ounces  to  scruples.  1368. 

21.  Bring  27  drams  to  grains.  1620. 

22.  Bring  480  scruples  to  pounds. 

Long  Measure, 

23.  Bring  40  yards  to  feet.  Ans,  120. 

24.  Bring  74  poles  to  feet.  1221. 

25.  Bring  120  furlongs  to  poles.  4800. 

26.  Bring  27  yards  to  inches.  972. 

27.  Bring  1  mile  to  yards. 

Cloth  Measure. 

28.  Bring  5  yards  to  quarters.  Ans.  20 

29.  Bring  10  yards  to  nails.  160. 

30.  Bring  72  quarters  to  nails.  288. 

31.  Bring  144  quarters  to  nails. 

Land  or  Square  Measure, 

32.  Bring  18  roods  to  poles.  Ans.  720 


SIMPLE    REDUCTION. 


69 


33.  Bring  45  acres  to  poles. 

34.  Bring  12  acres  to  poles. 

35.  Bring  27  roods  to  poles. 

Liquid  Measure, 

36.  Bring  32  gallons  to  pints. 

37.  Bring  4  hogshecfQs  to  quarts. 

38.  Bring  1  hogshead  to  gills. 

39.  Bring  1  barrel  to  pints. 

40.  Bring  504  pints  to  barrels. 

Dry  Measure, 

41.  Bring  14  bushels  to  pecks. 

42.  Bring  1664  quarts  to  bushels. 

43.  Bring  1152  pints  to  bushels. 

44.  Bring  84  bushels  to  quarts. 

45.  Bring  70  bushels  to  pints. 

Time, 

46.  Bring  7  weeks  to  hours. 

47.  Bring  2  days  to  minutes. 

48.  Bring  11  days  to  seconds. 

49.  Bring  32  weeks  to  hours. 

50.  Bring  47  years  to  days. 

51.  Bring  259200  seconds  to  days. 

Promiscuous  Questions, 

52.  Bring  15  years  to  weeks. 

53.  Bring  1  mile  to  inches. 

54.  Bring  97  leagues  to  poles. 

55.  Bring  88  degrees  to  miles. 

56.  Bring  4730  yards  to  inches. 

57.  Bring  400  pints  to  gallons. 

58.  Bring  6  hogsheads  to  quarts. 

59.  Bring  1800  minutes  to  hours. 

60.  Bring  50  yards  to  nails. 

61.  Bring  131bs.  avoirdupois  to  drams. 

62.  Bring  6784  drams  to  pounds. 

63.  Bring  1  ton  to  drams. 

64.  Bring  15  years  to  seconds. 

65.  Bring  1664  pints  to  bushels. 

66.  Bring  90  furlongs  to  yards. 

67.  Bring  47  acres  to  poles. 
68  Bring  48  poles  to  inches. 


Ans  7200 
1920. 
1080. 


Ans.  256. 

1008. 

2016. 

252. 

2. 


Ans.  56 

52. 

18. 

2688. 


Ans,  1176. 

2880. 

950400. 

5376. 

17155. 


Ans.  780. 

63360. 

93120. 

6116. 

170280. 

50. 

1512. 

30. 

800. 

3328. 

26.5. 

573440. 

473040000. 

26. 

19800. 

7520 

9504 


70 


PRACTICAL    EXAMPLES. 


69.  Bring  4  signs  to  seconds.  Ans,  432000 

70.  Bring  7  hogsheads  to  gallons. 

71.  Bring  27  barrels  to  pints. 

72.  Bring  8  miles  to  yards. 

73.  Bring  60  poles  to  inches. 

74.  Bring  1478  pints  to  gallons. 

75.  Bring  7644  inches  to  yards.  • 

76.  Bring  82800  seconds  to  hours. 

77.  Bring  2764  ounces  to  cwt. 

78.  Bring  87960  hours  to  weeks. 

79.  Bring  116424  cubic  inches  to  tons. 

80.  Bring  4014489600  square  inches  to  square  acres.^y  . 


SHORT  PRACTICAL  EXAMPLES, 

APPLICABLE     TO     REDUCTION     AND     DECIMALS. 

3.  What  cost  15lbs. 


I.  What  cost  4^1bs.  sugar 
at  12c.  per  pound? 
12 
4X 

48 
^Ib.  =6 


2 

per  lb.  ? 


54c.  Ans, 
What  cost  5|lbs.  at  14c. 


14 
5X 


70 


731c 


per  pound  1 


at  16^c 


16i 
15x 


2.40 


71-rr^of  15 


D2.47^  Ans, 
4.  What  cost  22jlbs.  at  lie 
per  pound  ? 
11 
22  X 


2.42 


8|  =  I  of  11 


5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


.  Ans.  D2.50i  Ans. 

5  yards  at  D2.25  per  yard  ?        Ans.  D  11.25. 


What  cost 

What  cost  6|  yards  at  D2.50  per  yard  ? 


What  cost 
What  cost 
What  cost 
What  cost 
What  cost 
What  cost 


13.  What  cost 


D16.25. 
51-  lbs.  at  12^c.  per  pound?  68|c. 

25|  lbs.  at  17c.  per  pound?  D4.37f 

7  galls.  3  qts.  at  62^c.  per  gall.  ?    D4.84.375. 
2  qts.  1  pt.  at  17jc.  per  quart?  43|c. 

2i  cwt.  at  D4.50  per  cwt.  ?  Dl  1 .25. 

5  cwt.  1  qr.  14  ]bs.  at  D5  per  cwt.  ?    D26.87.5 
6|  cwt.  7  lbs.  at  D4.50  per  cwt.  ?       D30.65 


1>RACTICAL    EXAMPLES.  71 

14.  What  cost  97  lbs.  at  75c.  per  lb.  ?  Ans,  D72.75 

15.  What  cost  49  lbs.  at  12lc.  per  lb.  ?  D6.121. 

,  16.  What  cost  42^  lbs.  at  ST^c.  per  lb.  ?  Dl5.93^*. 

17.  What  cost  6.87  lbs.  indigo  at  D2.25  per  lb.  ?  Dl5.45f. 

18.  What  cost  1 3.  bushels  at  D  1.25  per  bushel?  D2.18|. 

19.  What  cost  3  pecks  at  Dl. 50  per  bushel?  Dl.l2^ 

20.  What  cost  J-  bushel  at  Dl.37i  per  bushel  ?  .45.8. 

21.  What  cost  1  peck  clover-seed  at  D6.25  per  bush."?  Dl.56|-. 

22.  What  cost  2  galls.  1  pt.  at  D1.12^  per  gall.  ?  D2. 39.06. 

23.  What  cost  1  ^  tons  of  hay  at  D  17.50  per  ton  ?  D26.25. 

24.  What  will  1  cwt.  1  qr.  cost  at  6c.  per  lb.  ?  D8.40. 
^  25.  Wh^t  will  2  cwt.  17  lbs.  cost  at  6.5c.  per  lb.  ?  D15.66.5. 
:  26.  What  will  1  bush.  3  pecks  cost  at  7c.  per  qt.  ?  D3.92. 
[27.  What  will  1  cwt.  12  lbs.  cost  at  2^c.  per  oz.  ?,  D49.60. 
[28.  What  will  1  cwt.  cost  at  S^c.  per  dram  ?  D1003.52. 
^  29.  If  ^  of  a  yard  cost  ^  of  a  D.,  how  much  will  17  yds.  cost  ? 

_    30.  If  ^D.  will  pay  for  |  of  a  yard  of  riband,  how  many  yards 
can  you  buy  for  D4  ?  for  D12  ?  for  D20  ?  for  D26  ? 

REVIEW. 

What  things  are  weighed  by  Avoirdupois  Weight?  What 
are  the  denominations  ?  Repeat  the  table.  How  do  they 
weigh  in  the  principal  cities?  Ans.  By  the  100  lbs.  What 
is  the  use  of  Troy  Weight?  Repeat  the  table.  When  is 
Apothecaries'  Weight  used,  and  for  what  purpose  ?  When  is 
Long  Measure  used  ?  What  are  the  denominations  ?  Repeat 
the  tables,  &c.  For  what  is  Land  Measure  used  ?  What  is  a 
square  ?  Ans,  A  figure  bounded  by  four  equal  sides,  or  lines, 
and  whose  angles  are  all  equal,  or  right  angles.  What  is  a 
square  number?  Ans.  Any  number  multiplied  into  itself,  as 
4x4=^16,  &c.  How  can  you  find  the  number  of  small  squares 
contained  in  a  large  square  ?  What  chain  is  used  in  surveying 
land  ?  How  long  is  it  ?  How  is  land  generally  estimated  ? 
When  is  Solid  or  Cubic  Measure  used  ?  What  are  the  denom- 
inations ?  How  many  inches  in  a  cubic  foot  ?  How  many  feet 
in  a  cord  of  wood  ?  in  a  quarter  of  a  cord  ?  How  many  cubic 
inches  in  a  gallon  ?  What  are  the  dimensions  of  a  bushel  ? 
Time.  Repeat  the  table.  Which  of  the  months  have  30  days, 
and  which  31  days?  For  what  is  Circular  Measure  used? 
How  many  things  make  a  dozen  ?  How  many  make  a  score  . 
How  many  sheets  make  a  quire  of  paper  ?  How  many  quires 
3fiake  a  ream  ?  How  many  sheets  are  there  in  17  reams  ?  How 
many  kinds  of  Reduction  are  there  ?  What  are  they  called  ? 
For  what  purpose  is  Reduction  used  ?     What  is  Simple  Reduc- 


72  ADDITION    OF    DIFFERENT    DENOMINATIONS. 

tion?  "When  high  denominations  are  to  be  brought  to  lower 
denominations,  how  do  you  proceed?  What  is  this  kind  of 
Reduction  sometimes  called?  When  low  denominations  are  to 
to  be  brought  higher,  how  will  you  proceed  ?  What  is  this 
called  ?    What  is  the  rule  ?     How  can  you  prove  Reduction? 


ADDITION  OF  DIFFERENT  DENOMINATIONS. 

When  it  is  required  to  add  together  several  numbers  of  dif- 
ferent denominations,  such  as  pounds,  ounces,  &;c.,  for  the  pur- 
pose of  finding  the  sum,  or  amount  total,  it  has  generally  been 
designated  as  Compound  Addition;  or,  by  some.  Denominate 
Numbers^  which  signifies  *'  to  name,"  and  Denomination,  which 
signifies  "  a  name  given  to  a  thing,  title,  &c.,"  either  of  which 
may  be  correct,  as  it  is  unimportant  what  the  title  is,  provided 
the  subject  is  understood,  although  we  have  given  the  last-men- 
tioned the  preference.  The  Cwt.  avoirdupois  (112lbs.)  has 
been  in  use  many  years,  but  is  not  so  much  used  of  late,  as  the 
merchants  in  the  principal  cities  have  substituted  the  lOOlbs. 
net  weight  for  the  112lbs.  avoirdupois.  This  change  will  be 
attended  with  many'advantages,  it  being  by  far  the  most  agree- 
able and  expeditious  method  of  computation,  as  it  could  easily 
be  reduced  to  a  system  of  decimal  calculations,  and  the  division 
of  the  lOOlbs.  into  lOths  ;  and  this  system  should  extend  to 
LENGTH,  and  our  weight,  measure,  &:c.,  would  correspond  with 
our  currency,  which  may,  and  probably  will,  some  day  not  far 
distant,  be  the  case. 


1.  Write  down  the  numbers  so  that  each  denomination  may 
stand  directly  under  each  other,  leaving  a  small  space  between 
them. 

2.  Add  the  right-hand  denomination,  the  same  as  in  Simple 
Addition. 

3.  Then  divide  that  amount  by  as  many  as  it  requires  of  that 
denomination  to  make  one  in  the  next. 

4.  Set  down  the  remainder  under  that  denomination,  and 
carry  the  quotient  to  the  next  denomination,  and  add  it  in ;  if 
there  be  no  remainder,  set  down  a  cipher. 

5.  Continue  in  this  way  through  to  the  last  denomination, 


ADDITION    OF    DIFFERENT    DENOMINATIONS. 


7a 


which  add,  the  same  as  in  simple  addition,  and  set  down  the 
whole  amount.     Proof :  the  same  as  in  whole  numbers. 


EXAMPLES. 


10    20 

T.  cwt. 

2     3 

4     1 

2     4 


4.  28.  16.  16. 
qrs.  lbs.  oz.  dr. 

2  11     6     8 

3  16  11   12 
0     18  10  14 


8     9     2     18  13     2 


Explanation.  This  example  be* 
longs  to  avoirdupois  weight,  but  the 
same  general  rule  will  hold  good  in 
all  cases  of  compound  numbers.  The 
first  denomination  at  the  right  hand 
is  drams,  which  add,  the  same  as  in 
simple  addition,  and  the  amount  is 
34 ;  this  is  divided  by  1 6,  because  1 6 
drams  make  1  oz.,  and  we  have  2  ounces  in  the  quotient,  and 
2  drs.  over  ;  set  down  the  2  drs.  and  carry  the  2  oz.  to  the  de* 
nomination  of  that  name,  which  add  in,  and  it  will  make  29  oz. ; 
divide  the  29  oz.  by  16,  to  get  the  pounds,  and  it  will  give  1  lb. 
and  13  oz.  over ;  set  down  the  13  oz.,  and  carry  the  1  lb.  to  the 
next  denomination  ;  then  add  that  denomination,  and  it  will  give 
46  lbs.  ;  divide  the  lbs.  by  28,  to  bring  them  to  quarters,  and 
you  have  1  qr.  and  18  lbs. ;  set  down  the  18  lbs.,  and  carry  the 
1  qr.  to  the  denomination  of  that  name,  which  add  in,  and  you 
have  6  qrs.,  which  divide  by  4,  because  4  qrs.  make  1  cwt.,  anc^^ 
you  have  1  cwt.  and  2  qrs.  ;  set  down  the  2  qrs.,  and  carry  tho 
1  cwt.  to  the  next  denomination,  and  add  it  in ;  you  will  then 
have  9  cwt.,  there  being  none  to  carry  now,  because  it  is  less 
than  a  ton ;  then  add  and  set  down  all  of  the  next  denomina- 
tion, and  you  have  the  sum  total. 


1. 


lbs. 
478 
321 
214 
321 


oz. 

8 
4 
8 
4 


decimally. 


lbs. 
238 
342 
137 
320 


oz. 
6 
10 
4 
0 


1335 


8=1335.5  at  4c, 
4X 


1038     4  =  1038.25  at  5c. 
5X 


D.53.42.0 


D.51.91.25 


AVOIRDUPOIS    WEIGHT. 

T.cwt.qr.lb,  4.  cwt.qr.lb.  oz. 


3  19  3  27 

2  12  1  12 

10  5  0  00 

3  2  10 

T.17  00  3  21 


6  2  U 

3  1  14 

4  0  16 

5  0  10 
3  3  9 


TROY  WEIGHT. 

5.  lb.  OZ.  dwt.  6.  lb.  oz.  dwt.  gr. 


93  11  18 

6  0  1 

14  4  12 

72  11  3 

lbl87  3  14 


27  9 


20 
24 
18 
16 


16 
4 
11 
11 
18 


18 
11 
19 
16 
10 


74 


ADDITION    OF    DIFFERENT    DEXOMINATIO.^  . 


APOTHECARIES     WEIGHT. 

7.  Ib.oz.dr.scr.  8.  lb.oz.dr.scr.gr. 

6  3  12  21   6    4    1    13 

19  9  5   1  27  4    3    2    14 

182  7  3  2  34  2    2    1    15 

57  6  1   0  20  7   2    2    10 

40  000  7630    11 

10  3    4    1    18 

306  2  3  2  


LONG  MEASURE. 

9.  yd.  ft.  in.  10.  L.  m.  fur.  po. 

2  2  9  8  2  7  16 

110  20  1  6  17 

10  1  3  24  2  4  19 

12  0  4  50  0  2  11 

18  1   1   12 

26  2  4 


CLOTH    MEASURE. 

11.  yd.  qr.  na.  12.  EE.  qr.  na. 

46     2     1  86     4      2 

79     2     3  44     3      3 

25     2     2  21     2       1 

14     2     1  18     1       2 

20     4       1 

166    ^1     3 

LAND    OR    SQUARE    MEASURE. 

13.  yd.  ft.  in.  14.  A.  R,  po. 

8  2  12  150  3  39 

10  1  95  265  2  11 

12  1  115  284  1  12 

20  0  46  326  0  20 

171  3  18 

50  5  124    


LIQUID  MEASURE. 

15.  gall.  qt.  pt.  16.  hhd.  gall,  qt 

24     2     1  2       11       3 

14     1     0  4       16      2 

6     3     0  10       10       1 

8     2      1  11         9      0 

12       11       2 

54     1     0 


17.       DRV     MEASURE.       18. 

bush.  pk.  qt.  bush.  pk.  qt.  pt. 

10     3     2  36     1     7     1 

117     1     3  48     2     6     0 

215     2     4  71     2     4     1 

450     3     0  16     3     0     0 

18     3      5     1 

794     2     1        . 


19.      CUBIC    MEASURE.      20. 

T.  ft.    C.  ft.  ft.  in. 

41  43  3  122  13  1446 

12  43   4  114  16  1726 

49  6   7  83  3  866 

4  27  10  127  17  284 


108  19 


21.  TIME.  22. 
da.  h.  m.  sec.  w.  da.  h.  m. 
4  20  56  54  2  1  10  40 
3  19  25  22  14  11  27 
2  8  0  3  4  6  14  16 
0  6  0  0  6  3  16  18 
7  2  10  19 


11     6  22  19 


MOTION  OR  CIRCLE  MEASURE. 

23.  24. 

sig.  °    '  ''  sig.  "^     '    " 

1  5  7  32  2  7  32  29 

1  7  26  12  0  4  21  18 

4  8  26  11  1  5  20  16 

1  4  32  17  3  6  10  19 

3  6  0  47  4  3  11  17 

2  2  16  20 

11  1  32  59 • 

25. 
deg.  m.  fur.  po. 

94  65  6  38 

37  47  3  28 

50  34  1  12 
152  14  4  0 


26. 
Y.  m.  da. 
28  10  26 
34  11  17 
63  9  12 
21  3  7 


ADDITION    OF    DIFFEREx\T    DENOMINATIONS.  75 

27.  A  purchased  4-  hogsheads  of  sugar,  which  weighed  as 
follows  :  No.  1,  5  cwt.  3  qrs.  21  lbs.  ;  No.  2,  4  cwt.  2  qrs.  27 
lbs.  ;  No.  3,  7  cwt.  0  qrs.  1  8  lbs. ;  No.  4,  6  cwt.  2  qrs.  13  lbs. ; 
what  is  the  weight  of  them  all  ?  Ans.  24  cwt.  1  qr.  23  lbs. 

28.  A  man  bought  5  pieces  of  sheeting:  1st  piece,  35  yds. 
1  qr.  2  na. ;  2d,  40  yds.  2  na.  ;  3d,  46  yds.  3  qrs.  3  na.  ;  4th, 
50  yds.  1  qr.  3  na. ;  5th,  54  yds.  3  qrs.  3  na.  ;  required  the 
number  of  yards  in  the  5  pieces.        Ans.  227  yds.  3  qrs.  1  na. 

29.  A  farmer  disposed  of  4  bags  of  grain,  which  measured 
as  follows  :  1st,  3  bush.  2  pks.  6  qts. ;  2d,  2  bush.  2  pks.  7  qts. ; 
3d,  3  bush.  1  pk.  4  qts. ;  4th,  3  bush.  1  pk.  3  qts.  ;  how  much 
grain  did  he  sell?  Ans.  13  bush.  4  qts. 

30.  A  merchant  had  4  hogsheads  of  wine,  on  measuring 
which,  they  were  found  to  fall  short  of  the  quantity  purchased  : 
the  1st  contained  59  galls.  3  qts.  1  pt. ;  2d,  58  galls.  2  qts.  1  pt. ; 
3d,  60  galls.  1  qt.  1  pt.  ;  4th,  62  galls.  3  qts.  1  pt.  ;  how  many 
gallons  did  he  have  ?  A?is.  241  galls.  3  qts. 

31.  C.  has  five  fields  :  the  1st  contains  25  A.  1  R.  30  po.;  2d, 
28  A.  3  R.  18  po.  ;  3d,  30  A.  1  R.  27  po. ;  4th,  34  A.  20  po. ; 
5th,  35  A.  2  R.  24  po.  ;  how  many  acres  in  the  five  fields  1 

Ans.  154  A.  1  R.'39  po. 

32.  Add  172  years,  1  week,  4  hours,  52  sec. ;  34  m.  18  sec. ; 
15  y.  4  mo.  5da.  3  h.  27  m.  ;  1  w.  3  da.  21  h.  35  m.  18.  sec. 
together.  Ans.  187  y.  4  mo.  3  w.  2  da.  5  h.  37.  m.  28  sec. 

33.  Add  19  T.  2  hhds.  19  galls. ;  45  T.  1  qt.  1  pt. ;  3  hhds. 
17  galls.  2  qts.;  21  galls.  1  pt.  together. 

Ans.  65  T.  1  hhd.  58  galls.  0  qt.  0  pt. 

34.  A  tailor  bought  4  pieces  of  broadcloth  :  the  1st  piece 
contained  25  yds.  2  qrs.  2  na. ;  2d,  21  yds.  0  qrs.  2  na. ;  3d, 
26  yds.  1  qr. ;  4th,  27  yds.  ;  for  the  1st  piece  he  paid  D85.50  ; 
2d,  D73.40;  3d,  D90.10;  4th,  D91.00;  how  many  yards  did 
he  have  ?  how  much  did  it  all  cost  ?  and  how  much  per  yard  1 

A?is.  100  yds. ;  cost  in  all,  D340  ;  price  per  yard,  D3.40. 

35.  Add  27.25  yds. ;  16.75  yds.  ;  18.5  yds.  together,  and 
give  the  value  at  D4.25  per  yd. 

36.  Add  106.6  pounds;  218.25  pounds;  374.18  pounds; 
97.75  pounds  together,  and  give  the  value  at  7.5c.  per  pound. 

37.  Add  56.16  acres  ;  97.75  acres  ;  84.62  acres  ;  45.21  acres 
together,  and  givethe  value,  alD50. 50  per  acre.  Ans.  D14328.87. 

REVIEW. 

What  is  the  use  of  Addition  of  Different  Denominations  ? 
What  is  a  denominate  number  ?  How  do  you  set  down  the 
numbers  in  this  rule  ?     Where  do  you  begin  to  add  ?     How  do 


76  SUBTRACTION    OF    DIFFERKNT    DENOMINATIONS. 

you  proceed  after  placing  tlie  denominations  under  each  other ' 
By  what  do  you  divide  the  amount  of  a  column,  or  denomina- 
tion ?  If,  after  dividing,  there  is  any  remainder,  what  will  you 
do  with  it  ?  and  how  do  you  proceed  if  there  be  no  remainder  ? 
What  will  you  do  with  the  quotient  ?     Proof. 

38.  Add  together  20  yrs.  3G3  da.  20h.  50m.  30  sec.  ;  20 
yrs.  40  da.  10  h.  30  m.  20  sec;  12  yrs.  110  da.  13  h.  16  sec; 
13  yrs.  8  da.  10  h.  20  m.  14  sec;  7  yrs.  20  da.  8  h.  10  m.  12 
sec  A?is.  73  yrs.  178  da.  14h.  51  m.  32  sec 


SUBTRACTION  OF  DIFFERENT  DENOMINATIONS. 

This  rule  is  used  when  numbers  of  different  denominations 
are  given  to  be  subtracted,  or  a  smaller  number  from  a  greater 
of  a  like  denomination,  and  show  their  difference  or  remainder. 


1.  Write  down  the  larger  number,  and  directly  under  it  the 
less  number,  so  that  the  same  denominations  shall  stand  under 
each  other. 

2.  Begin  at  tlie  right  hand,  and  subtract  the  lower  from  the 
upper  number,  if  that  be  the  largest,  and  set  down  the  remainder. 

3.  But  if  the  lower  number  is  more  than  the  one  above  it, 
then  subtract  from  as  many  as  it  takes  of  that  denomination  to 
make  one  in  the  next ;  take  the  difference,  and  add  it  to  the 
upper  number,  set  it  down  ;  then  carry  one,  and  add  it  to  the 
next  lower  denomination,  and  continue  through  the  sum  in  this 
manner,  and  in  the  last  denomination  subtract  the  same,  as  in 
simple  subtraction.  Proof  the  same  as  in  simple  subtraction ; 
observing  to  carry  as  above  directed. 

EXAMPLES    AND    QUESTIONS. 


10. 

100. 

12. 

4. 

7. 

24. 

60. 

60. 

Cent. 

yrs. 

mo. 

w. 

da. 

h. 

m. 

sec 

7 

97 

7 

3 

2 

3 

20 

30 

5 

98 

9 

0 

3 

5 

50 

40 

98       10         2         5       21         29         50 

97         7         3         2         3         20         30  proof 


SUBTRACTION    OF    DIFFERENT    DENOMINATIONS. 


77 


Explanation. — Place  the  numbers  over  their  respective  de- 
nominations, as  in  Compound  Addition ;  then  begin  at  the  right 
hand  with  40  ;  40  from  60  will  leave  20,  which  added  to  30 
will  make  50,  which  set  down,  and  carry  1  to  50  =  51  from  60, 
9  will  remain,  add  this  to  20=29,  which  set  down  ;  now  carry 
1  to  5  =  6  from  24  will  leave  18,  and  3  are  21  ;  then  1  to  carry 
to  3  is  4,  4  from  7  will  leave  3,  and  2  are  5 ;  then  1  to  carry  to 

0  is  1,  1  from  3  and  2  remain ;  9  from  12  and  3  remain,  which 
added  to  7  are  10 ;   then  1  to  carry  to  98  is  99 ;  99  from  100, 

1  remains,  which  added  to  97  makes  98  ;  then  1  to  carry  to  5  is 
6,  6  from  7  and  1  remains,  whiah  set  down.  For  the  proof, 
add  and  divide  as  in  the  example. 


1.       AVOIRDUPOIS    WEIGHT.        2. 

T.  cwt.qrs.lb.  T.cwt.qr.lb.oz. 
52  12  3  15   24  18  3  12  9 
24  10  0  26   20  19  2  18  2 


28  2  2  17 

3.       TROY  WEIGHT.       4. 

lb.oz.dwt.gr.   lb.oz.dwt.gr. 


45  9 
15  6 


4  3 
18  17 


21  7  14  1 
16  9  18  12 


30  2  5  10 

5.  apothecaries'  weight.  6. 
lb.  oz.  dr.  scr.  lb.  oz.  dr.  scr. 
49  8  6  2   84  7  5  2 
21  3  5  1   75  8  6  1 


9.             CLOTH  MEASURE.           10. 

yds.  qr.  na.  yds.  qr.  na. 

784     2     1  64     2     3 

651     3     2  18     3     0 


11.             LAND    MEASURE.  12. 

A.  R.  po.      A.  R.  po. 

540  2  20     764  2  13 

332  0  25     684  3  17 

7* 


13.  LIQUID    MEASUDE.  14. 

T. hd.gal.qt.pt.  T.hd.gal.qt  pt. 
2    2  40    1    1    27  1   18    2    1 
1    1   14    3    1    18  2  20    3    1 


1  1 

25 

2 

0 

15. 

DRY 

MEASURE. 

16. 

bush 

pk. 

qt.  pt. 

bush 

pk.  qt.pt. 

220 

2 

0 

1 

643 

2 

6  0 

214 

1 

1 

1 

324 

1 

7  1 

6     0     7     0 

17.  CUBIC    MEASURE.  18. 

T.  fh.  C.  ft.    T.  ft.  in. 

116  24  72  114   45  18  140 

109  39  41  120   16  14  145 


28 
7. 
L. 
56 
10 

5 

m. 

1 
0 

1  1 

LONG  MEASURE.      8. 

fur.  po.  fur  po.  yd.ft.in. 
0  19  29  6  3  2  7 
7  20  18  7  4  2  8 

6  25 
19.         TIME.         20, 

Y.  m.  w.  d.  h.  mi.  Y.  m.  w.  d.  h. 
27  3  2  4  18  40  28  9  2  4  16 
13  2  1  5  19  20  24  10  3  5  18 

46 

0 

0  39 

14  1  0  5  23  20 

MOTION    OR    CIRCLE    MEASURE. 

21.sig.  **  '   ''22.sig.°  '    " 

10  2  3  20   24  7  42  27 

4  8  20  30   18  9  14  28 


5  23  42  50 


23. 

Y.mo.  w.  d.  ho.  mi.  sec.  dec. 

6  0  3  1  3  40  20  .5 

1  0  2  6  2  57  36  .2f 


78  SUBTRACTION    OF    DII-FKRENT    DENOMINATIONS. 

24.  W.  has  a  cask  of  sugar  weighing  7  cwt.  2  qrs.  18  lbs.  12 
oz.  ;  if  he  shoiilcl  sell  3  cwt.  2  qrs.  20  lbs.,  how  much  would  he 
have  remaining?  Ans.  3  cwt.  3  qrs.  26  lbs.  12  oz. 

25.  C.  had  35  yds.  2  qrs.  1  na.  of  sheeting  ;  he  sold  two 
pieces,  one  of  7  yards.  1  qr.  2  na.,  the  other  1 1  yds.  2  qrs.  1  na. ; 
how  many  yards  remain  of  the  piece  ?     Ans.  1 6  yds.  2  qrs.  2  na. 

26.  A  farmer  has  3  farms;  the  first  contains  321  A.  2  R.  10 
no.,  the  second  231  A.  18  po.,  the  third  180  A.  0  R.  14  po. ; 
if  he  should  sell  274  A.  1  R.  19  po.,  how  many  acres  would  ho 
have  remaining  ?  Ans.  458  A.  1  R.  23  po. 

27.  W.  purchased  three  hogsheads  of  wine  ;  if  he  should 
sell  127  gallons,  3  quarts,  1  pint,  how  many  gallons  would  re- 
main ?  Ans.  61  gallons,  1  pint. 

28.  Subtract  500  bushels,  1  peck,  4  quarts,  from  740  bush- 
els, 3  quarts.  Ans.  239  bushels,  2  pecks,  7  quarts. 

29.  If  it  be  415  miles,  3  furlongs,  20  poles,  to  Pittsburgh,  and 
375  miles,  1  furlong,  30  poles,  to  Baltimore,  how  much  farther 
is  it  to  Pittsburgh  than  to  Baltimore  ?     Ans.  40  m.  1  fur.  30  po. 

30.  From  76  years  take  27  years,  9  months,  3  weeks,  2  days. 

Ans.  48  years,  2  months,  0  weeks,  5  days. 

To  find  the  difference  between  two  given  dates. 


Write  down  the  larger  number,  and  under  it  the  less  number. 
\{  the  number  of  days  in  the  less  number  be  more  than  in  the 
greater,  subtract  from  as  many  as  there  be  days  in  the  month 
mentioned  in  the  less  number,  and  add  in  the  days  in  the  greater 
number,  which  set  down,  and  carry  one  to  the  month,  then  sub 
tract  in  the  usual  way. 

31.  W.  was  born  the  18th  of  the  4th  month,  1775  ;  how  old 
was  he  on  the  25th  of  the  3d  month,  1824 1 
Year,    month,  day. 

Thus:  1824         3         25 

1775         4         18 


48       11  7  Ans. 

32.  A  person  was  born  March  17,  1796  ;  required  his  agu 
August  24,  1824.  Ans.  28  years,  5  months,  7  days. 

33.  A.  had  500  bushels  of  wheat;  *^he  sold  to  B.  150.75,  to 
C.  62.25,  to  D.  18.5,  for  Dl.75  per  bushel;  how  many  bushels 
had  he  remaining,  and  how  much  money  did  he  receive  ? 

34.  A  merchant  had  3  hogsheads  of  sugar,  which  weighed 
2764.5  pounds  ;  he  sold  1421.25  pounds  for  DUO;  how  much 


MULTIPLICATION    OF    DIFFERENT    DENOMINATIONS.  79 

did  he  receive  per  pound  ?  and  how  much  would  the  balance  he 
has  remaining  come  to,  at  the  same  price  ? 

35.  Take  252.76  acres  from  431.18  acres,  and  calculate  the 
value  of  the  remainder  at  D67.27  per  acre  ? 

36.  There  are  two  men,  the  oldest  is  81  years,  6  months,  3 
weeks,  1  day,  21  hours,  16  seconds  ;  the  youngest  29  years,  10 
months,  2  weeks,  4  days,  16  hours,  34  minutes,  45  seconds; 
what  is  the  difference  of  their  ages  ? 

37.  What  is  the  difference  of  time  between  31  years,  10 
months,  2  weeks,  4  days,  7  hours,  24  minutes,  49  seconds,  and 
10  years,  10  months,  2  weeks,  2  days,  7  hours,  59  minutes,  14 
seconds?  Ans.  21  years,  1  day,  23  hours,  25  min.  35  sec. 

38.  A  merchant  bought  375  tons,  15  cwt.  3  quarters,  19  lbs. 
7  oz.  12  dr.  of  sugar,  and  sold  205  tons,  17  cwt.  1  quarter,  27 
lbs.  9  oz.  15  dr.  ;  how  much  had  he  remaining  1 

Ans.  169  tons,  18  cwt.  1  qr.  19  lbs.  13  oz.  13  dr. 

39.  Bought  a  piece  of  cloth  containing  145  yards,  3  quarters, 
and  sold  95  yards,  2  quarters,  3  nails,  how  much  remains  ? 

Ans.  50  yards,  1  nail. 

40.  From  174  hhds.  10  galls.  1  qt.  1  pt.  take  86  hhds.  17 
galls.  2  qts.  1  pt.  Ans.  87  hhds.  55  galls.  3  qts 

REVIEW. 

What  do  you  understand  by  Subtraction  of  Different  Denom- 
inations, and  when  is  it  used?  How  do  you  write  down 
numbers  in  this  rule  ?  What  do  you  place  over  the  several  de- 
nominations ?  Where  do  you  begin  to  subtract  ?  When  the 
number  to  be  subtracted  is  less  than  the  one  above  it,  what  will 
you  do  ?  When  it  is  greater,  what  will  you  do  ?  What  is  the 
rule  ?  How  do  you  prove  questions  in  this  rule  ?  How  do  you 
set  down  two  given  calendar  dates,  in  order  to  find  the  difference 
between  them  ?  After  having  set  down  the  two  given  dates, 
where  do  you  begin  to  subtract  ?  How  do  you  proceed  when 
the  number  of  days  in  the  less  date  is  greater  than  the  number 
of  days  in  the  greater  date  ?     What  is  the  rule  ? 


MULTIPLICATION  OF  DIFFERENT  DENOMINA- 
TIONS. 

The  object  of  this  rule  is  to  perform  a  number  of  additions 
of  different  denominations,  or  when  numbers  of  that  description 
are  to  be  multiplied. 


80  MULTIPLICATION    OF    DIFFERENT    DENOMINATIONS. 

When  the  multiplier  does  not  exceed  twelve. 


Write  down  the  multiplicand,  and  write  the  quantity  of  the 
several  denominations  over  each,  as  directed  in  subtraction. 
Then  write  the  multiplier  under  the  lowest  denomination,  at  the 
right  hand ;  then  multiply  that  denomination,  and  divide  it  by 
as  many  as  it  takes  of  that  denomination  to  make  one  in  the 
next,  and  set  down  the  remainder,  and  carry  the  quotient  to  the 
product  of  the  next  denomination,  &c.  Proof  as  in  simple 
multiplication. 


EXAMPLES. 


10.  20.  4.  28.  16.  16; 

T.  cwt.  qr.  lb.    oz.    dr. 

4     9      2    17    12    13 

7 


31     7      2    12      9    11 


Explanation.  First  7x13  =  91 
—  16  =  5  and  11  over;  now  12x7 
=  84  and  5  are  89-f-16=5  and  9 
over,  set  down  the  9  ;  then  17x7 
=  119  and  5  are  124^28=4  and 
12  over,  set  down  the  12  ;  2x7  = 
14  and  4  are  18-^4  are  4  cwt.  and 

2  qrs.  over,  set  down  the  2  qrs. ;  now  9  X  7  =  63  and  the  4  cwt. 

make  67-:- 20  =  3  tons  and  7  cwt.  over,  set  down  the  7  cwt. ; 

now  4x7=28  and  the  3  tons  are  31  tons. 

1.  2. 

lb.  oz.  dwt.  gr.  L.  m.  fur.  po. 

17     5     12     6  15     2     7    30 

3  6 


52     4     16  18     95     2     6    20 


bush.     pk.     qt. 

14         3         2 

6 


A.     R.    po. 
47     3      15 

2 

95     2      30 

5.  6. 

yr.  da.  h.  m.  sec.  lb.  oz.  pwt.  gr. 
4  5  20  32   10     4     1    15    22 

7  11 


88 


33  5  23  45  10   45     7    15     2 


7. 

8. 

yds.  ft.  in. 

yr.  mo.  w.  da. 

4     2     7 

7     8      3      6 

6 

9 

9.  10. 

yr.  da.  h.  m.  sec.  L.m.fur.po.yd. 
8  125  18  47  50    47  1   1     9     4 
4  5 


11.  12. 

lb.oz.dr.scr.gr.  lb.oz.dr.scr.gr 

19  10  6  1   15  26  11  5    2    16 

8  11 


13.  14. 

deg.  m.  fur.  po.  yd.    ft.  in. 

4     62     7     34  184     2  11 

7  9 


MULTIPLICATION    OF    DIFFERENT    DENOMINATIONS. 


8. 


15.  ^  IG. 

T.  cvvt.  qr.  lb.  T.  cwt.  qr.  lb.  oz. 

18  14    2   18  27    4     1    10     7 
9  8 


17.  18. 

mo.  w.  da.  h.  m.  sec.  yr.  m.da. 
8    3    6  215122     15  11  22.5 
5  11 


19.  Multipl)'  24  bushels  1  peck,  by  5, 

20.  Multiply  18  yds.  1  qr.  2  na.  by  7. 


Ans.  121  bush.  1  pk. 
128  yds.  2  qrs.  2  na. 


When  the  multiplier  exceeds  twelve^  and  is  the  product  of  any  two 
figures  in  the  multiplication  table. 

RULE. 

Multiply  the  sum  by  one  of  the  figures,  and  that  product  by 
the  other,  and  the  last  product  will  be  the  answer. 

21.  Multiply  20  yds.  2  qrs.  1  na.  by  42.     6x7=42. 

A?is.  863  yds.  2  qrs.  2  na. 

22.  Multiply  3  A.  2  R.  10  po.  by  49.  174  A.  2  R.  10  po. 

23.  Multiply  47  A.  3  R.  20  po.  by  54.      2585  A.  1  R.  0  po. 

24.  Multiply  48  M.  7  fur.  25  po.  by  88.     4307  M.  7  fur.  0  po. 

25.  Multiply  56  lb.  9  oz.  6 dr.  by  84.(apoth.)  47721b.  3  oz.  0  dr. 

26.  What  will  176.75  acres  come  to,  at  1)47.62  per  acre  ? 

27.  What  will  67.47  yards  of  cloth  cost,  at  D3.75  per  yard  ? 

28.  What  will  47.5  bushels  of  grain  cost,  at  Dl.25  per  bush.  ? 

29.  In  6  parcels  of  wood,  each  containing  5  cords,  96  feet, 
how  many  cords  1  Ans.  34^,  or  34,5. 

When    the    rnultiplier    exceeds    twelve^    and  is  not  a  composite 
number,  that  isy  the  product  of  two  figures, 

RULE. 

Multiply  the  simple  numbers  by  each  of  the  denominations 
separately,  and  reduce  each  product  to  the  highest  denomination 
named.  Then  add  the  several  products  together,  and  their  sum 
will  be  the  answer  sought. 

30.  Multiply  7  cwt.  3  qrs.  22  lbs. by  51 .  Ans.  405  cwt.l  qr.  2  lb. 

31.  Multiply  12  lb.  5  oz.  8  dwt.  by  39.     485  lb.  6  oz.  12  dwt. 

32.  Multiply  4  M.  6  fur.  21  po.  by  87.       418  M.  7  fur.  27  po. 

33.  Multiply25M.3fur.l8po.byl265.  32170  M. 4 fur.  10  po. 

34.  Multiply  48  ft.  4  in.  2  be.  by  2587.     125182  ft.  0  in.  2  be. 

35.  Multiply  4  hhd.  37  gall.  2  qt.  by  4250.     1 9529  hhd. 48 gall. 

36.  If  1  cord  of  wood  cost  D5.62^,  what  will  12  cords  cost  ? 

Ans.  67.47, 


82  DIVISION    OF    DIFFERENT    DENOMINATIONS. 

37.  What  costs  a  box  of  sugar,  weighing  106  pounds,  a 
151  c.  per  pound  ?  Ans.  Dl6.16f 

38.  At  D9.10  per  day,  how  much  a  year  ?  D3321.50. 


What  is  Multiplication  of  Different  Denominations  ?  When 
IS  it  used  1  Under  what  part  of  the  number  to  be  multiplied  do 
you  place  the  multiplier  ?  How  will  you  then  proceed  ?  By 
Avhat  do  you  di^ade  the  product  ?  If  there  is  a  remainder,  what 
is  to  be  done  ?  How  do  you  carry  ?  If  there  be  no  remainder 
after  the  division,  what  will  you  then  do?  What  will  you  do 
with  the  quotient  ?  Proof?  Repeat  the  rule.  Repeat  the  rule 
when  the  multiplier  is  the  exact  product  of  any  two  figures  in 
the  multiplication  table.  When  the  multiplier  is  not  a  compos- 
ite number,  and  more  than  twelve,  how  will  you  perform  the 
operation  ?     Repeat  this  rule. 


DIVISION  OF  DIFFERENT  DENOMINATIONS. 

By  this  rule  we  can  divide  a  number  containing  different  de- 
nominations. 


If  the  divisor  does  not  exceed  twelve,  proceed  as  in  short 
division  of  whole  numbers  ;  only  observe  this  difference,  that  in 
this  rule  there  are  several  denominations ;  therefore  divide  the 
left  hand  denomination,  and  place  the  quotient  under  it ;  reduce 
the  remainder,  if  any,  to  the  next  less  denomination,  and  add  in 
the  given  number  of  that  denomination  ;  then  proceed  as  before. 

2.  When  the  divisor  exceeds  twelve,  it  will  be  more  conve- 
nient to  work  by  long  division,  observing  the  above  rule.  Proof; 
by  multiplication. 

EXAMPLES. 

Divide  10  yards,  3  quarters,  2  nails,  among  5  persons. 

4.     4.  Explanation.     First  say,  5   in   10 

yds.qrs.na.  twice  and  0  over  ;  then  5  in  3  qrs.  0 

5)10     3     2  times,  3x4  =  12+2=14;  then  5  in 

1 4, 2  times  and  4  remainder  ;  and  the 

2     0     2-f  4  sum  is  done. 


DIVISION    OF    DIFFERENT    DENOMINATIONS. 


83 


3  If  Ans, 


4.  28.  16.  16. 

cwt.  qr.    lb.  oz.  dr.  cwt.  qr.  lb.  oz.  drams. 

40)50     2     14  9     11(1      1      1     12 

40  Explanation. — The  first  denomination  into 

—  which  you  divide  is  cwt. ;  here  you  have 
10  1  cwt.  and  10  cwt.  over,  or  remainder,  mul- 

4  X  tiply  this  remainder  by  4  qr.  and  add  in  the 

—  2  qr.  in  the  dividend,  and  you  have  42  qr. ; 
40)42(1  qr.  divide  as  before,  and  you  have  1  qr.  and  2 

40  qr.  over,  multiply  this  remainder  by  28  lb. 

—  and  add  in  the  14  lb.,  and  you  have  70  lb. ; 
2  divide  as  before,  and  you  have  1  lb.  and  30 

28  X  lb.  remaining,  multiply  the  30  lb.  by  16  oz. 

—  and  add  in  9  oz.,  and  you  have  489  oz. ;   di- 
40)70(1  lb.  vide  by  40  and  you  have  12  oz.  and  9  re- 

40  maining,   multiply  the  9  oz.  by  16  dr.  and 

—  add  in  the  11  dr.,  and  you  have  155  dr.  ; 
30  divide,  and  you  have  3  dr.  and  35  remain- 
16  X  der:  these  collectively  are  the  answer. 


40)489(12  oz. 
40 

89 
80 

9 
16x 

40)155(3  dr. 
120 


EXAMPLES. 


1.     cwt.  qr.   lb.         2.     yd.  *qr.  na. 
5)7     2    .15  7)25     2     1 


1     2       3 

3.     M.  fur.  po. 
6)45     7     18 


3     2     2+3 

4.     A.   R.  po. 
9)35     2     16 


7     5       9  +  4 


3     3     32+8 


rem.  35 

7. 
L.M.fur. 
12)62  2  18 

9. 
cwt.  qr.  lb. 
5)45     3     27 

11. 
yds.  qr.  na. 
7)44     1     2 


5.     yd.  qr.  na.     6.     A.   R.    po. 
10)87     2     3         7)59     2     37 


Y.  da.  h.  m.  sec 

6)47  96  18  47  55 

10. 

cwt.    qr.   lb. 

9)10     0     15 

12. 

yds.  qr.  na. 

11)56     3     3 


22 


I         13. 

M.  fur.  po. 
12)105     5 
15, 
hhd.  gall,  qt 
63)44   28    2 
17. 
D.       c. 
78)196     75 


14. 
M.     fur.  po. 
6)45     7     18 

16. 

hhd.  gall.  qt. 

120)150    47    3 

18. 

D.      c.  m. 

97)496     87     5 


84  DIVISION    OF    DIFFERENT    DENOMINATIONS. 

QUESTIONS. 

19.  Divide  482  bushels,  2  pecks,  4  quarts,  among  3  persons. 

Ans.  160  bushels,  3  pecks,  4  quarts,  each. 

20.  A  gentleman  bequeathed  to  his  5  sons  a  tract  of  land  con- 
taining 842  acres,  2  roods,  20  poles, to  be  divided  equally  among 
them;  required  the  share  of  each.  Ans.  168A.  2R.  4po. 

21.  Divide  17L.  IM.  4fur.  21po.  by  21.     Ans,  2M.  4fur.  Ipo. 

22.  Divide  4.5  gallons  of  wine  equally  among  144  persons. 

Ans.  1  gill  each. 

23.  From  a  piece  of  cloth  containing  64  yards,  2  nails,  a  tai- 
lor wishes  to  make  9  coats,  which  will  take  one  third  of  the 
whole  piece  ;  how  many  yards  did  each  coat  contain  ? 

Ans.  2  yards,  1  quarter,  2  nails. 

24.  If  90  hogsheads  weigh  56  tons,  13  cwt.  3  quarters,  10  lb., 
what  is  the  weight  of  one  hogshead  ?      Ans.  12  cwt.  2  qr.  11  lbs. 

25.  If  the  earth  revolve  on  its  axis  15  degrees  in  one  hour, 
how  far  does  it  revolve  in  one  minute  ?  Ans.  15'. 

26.  If  59  casks  contain  44  hhds.  53  gallons,  2  quarts,  1  pint, 
what  are  the  contents  of  each  cask  ? 

27.  When  175  galls.  2  quarts  of  beer  are  drunk  in  52  weeks, 
how  much  is  consumed  in  1  week?        Ans.  3  galls.  1  qt.  1  pt. 

28.  Divide  168  bushels,  1  peck,  6  quarts  of  wheat  among  35 
men. 

29.  What  will  be  the  share  of  one  man,  if  810  tons,  11  cwt 
20  lb.  10  oz.  11  dr.  be  divided  equally  among  346  men? 

Ans.  2  tons,  6  cwt.  3  qrs.  11  lb.  8  oz.  13  dr 

30.  Suppose  a  man  has  246  miles,  6  furlongs,  36  poles,  to 
travel  in  12  days,  how  far  will  that  be  in  a  day  ? 

Ans.  20  miles,  4  fur.  23  po. 

31.  If  a  steamboat  should  go  224  miles  a  day,  how  long 
would  it  take  her  to  go  to  China,  it  being  12000  miles  ? 

Ans.  53  days,  13  hours,  42  min.  51  sec. 

32.  If  one  man  can  lift  201  pounds,  12  ounces,  how  much 
can  a  boy  lift,  if  a  man  can  lift  8  times  as  much  as  a  boy  ? 

Ans.  25  lbs.  3  oz.  8  dms. 

33.  If  15  loads  of  hay  contain  35  tons,  4  cwt.,  what  is  the 
eight  of  each  load  ? 

34.  Divide  371  bushels,  1  peck  of  wheat  equally  among  270 
men.  Ans.  1  bushel,  1  peck,  4  quarts. 

35  Divide  19  cwt.  3  qrs.  27  lb.  12  oz.  of  sugar  among  48 
families.  Ans.  I  qr.  18  lb.  10  oz.+28. 

36.  Divide  4  cwt.  2  qrs.  7  lbs.  12  oz.  of  rice  among  14  per- 
sons. Ans.  1  qr.  8  lbs,  8  oz  +13 


APPLICATIOX    OF    THE    COMPOUND    RULES.  85 

37.  Divide  19  bushels,  2  pecks,  7  quarts,  1  pint,  2  gills. 
among  fi2  persons. 

38.  Divide  1749  acres,  2  roods,  30  poles,  equally  among  9 
men  and  7  women. 

REVIEW. 

When  is  Division  of  Different  Denominations  used  ?  Where 
will  you  place  the  divisor?  If,  when  you  divide  the  highest 
denomination  by  the  divisor,  a  remainder  occurs,  how  do  you 
proceed  ?  When  the  divisor  exceeds  twelve,  but  is  a  composite 
numbers,  how  can  the  operation  be  performed  1  Ans.  First 
divide  the  given  sum  by  one  of  the  numbers,  and  that  quotient 
by  the  other,  and  the  last  quotient  will  be  the  answer.  When 
the  divisor  exceeds  twelve,  and  is  not  a  composite  number,  how 
will  you  proceed?  Repeat  the  rule.  How  can  you  prove 
questions  in  this  rule  ? 


APPLICATION  OF  THE  COMPOUND  RULES. 

1.  A  man  travelled  in  one  day  27  miles,  3  furlongs  ;  another 
day,  30  miles,  2  furlongs,  25  poles  ;  required  the  distance. 

Ans.  57  miles,  5  furlongs,  25  poles. 

2.  A  man  has  three  farms,  the  first  contains  150  acres,  2 
roods,  25  poles;  the  second,  200  acres,  1  rood,  15  poles;  the 
third,  100  acres,  1  rood,  10  poles ;  how  many  acres  in  all^ 

Ans.  451  acres,  1  rood,  10  poles. 

3.  Add  2  cwt.  3  quarters,  27  pounds;  1  cwt.  2  quarters,  16 
pounds;  3  cwt.  1  quarter,  25  pounds;  5  cwt.  2  quarters,  12 
pounds;  2  cwt.  2  quarters,  14  pounds;  5  cwt.  1  quarter,  15 
pounds.  Ans.  21  cwt.  2  qrs.  25  lbs. 

4.  Bought  3  gallons,  2  quarts  of  wine  ;  5  gallons,  3  quarts  of 
vinegar ;  4  gallons,  1  qt.  of  molasses  ;  how  many  gallons  in  all  ? 

5.  A  tailor  purchased  a  piece  of  broadcloth  containing  40 
yards,  of  which  he  sold  36  yards,  1  quarter,  2  nails  ;  how  much 
will  he  have  remaining  ?  Ans.  3  yards,  2  quarters,  2  nails. 

6.  If  a  cask  of  wine  contains  54  gallons,  2  quarts,  1  piht,  2 
gills,  and  there  should  be  10  gallons,  3  quarts,  1  pint,  1  gill  sold, 
how  much  would  remain  ?       ^4;?^.  43  galls.  3  qts.  0  pts.  1  gilL 

7.  From  40  bush,  take  23  bush.  2  pks.    Ans.  16  bush.  2  pks. 

8.  From  1  square  yard  take  3.5  square  feet. 

9.  From  1  lb.  troy  take  15  grains.       Ans.  lloz.  19dwt.  9gr. 

10.  From  1  gall,  take  1  gill.  Ans.  3  qts.  1  pt,  3  gills. 

8 


86  APPLICATION    OF    THE    COMPOUND    RULES. 

11  A  merchant  bought  5  chests  of  tea,  each  weighin/^  ^cwt. 
2  qrs.  9  lbs. ;  what  was  the  weight  of  the  whole  ? 

Ans.  17  cwt.  3  qrs.  17  lbs. 
12.  From  1  acre  take  5^  poles.  Ans.  3  R.  34^  po. 

.13.  A  person  purchased  6  hhds.  of  sugar,  each  weighing  8 
cwt.  1  qr.  18  lbs. ;  required  the  weight  of  the  whole. 

Ans.  50  cwt.  1  qr.  24  lbs. 

14.  A  merchant  purchased  8  casks  of  oil,  each  containing  41 
galls.  3  qts.  1  pt. ;  required  the  number  of  gallons.        Ans.  335 

15.  In  28  pieces  of  calico,  each  containing  35  yards,  2  quar- 
ters, 1  nail,  how  many  yards  in  all  ?  Ans.  995  yds.  3  qrs. 

16.  A  farmer  has  6  bins  of  wheat,  containing  53  bushels,  3 
pecks,  5  quarts,  1  pint  each ;  how  much  in  all  1 

Ans.  323  bushels,  2  pecks,  1  quart. 

17.  Add  3  quarts,  3  pints,  1  gill ;  5  quarts,  4  pints,  3  gills , 
6  quarts,  1  gallon,  1  pint,  together. 

18.  From  1  ton  of  round  timber  take  50  cubic  inches. 

19.  From  1  year  take  12  hours.        Ans.  llmo.  3w.  6da.  12h. 

20.  From  12  miles,  15  rods,  take  3  furlongs. 

Ans.  11  m.  5  fur.  15  rods. 

PROBLEM  : 

Resulting  from  a  comparison  of  the  compound  rules,  by  the 
operation  of  which  many  questions  may  be  solved  in  a  short 
and  elegant  manner. 

1.  Having  the  sum  of  two  numbers,  and  one  of  them  given 
to  find  the  other. 

Rule. — Subtract  the  given  number  from  the  given  sum,  and 
the  remainder  will  be  the  number  required. 

Let  154  be  the  sum  of  two  numbers,  one  of  which  is  84,  the 
other  is  required.       154  sum— -84  given  number =70  the  other. 

2.  Having  the  greater  of  two  numbers,  and  the  difference 
between  that  and  the  less  given,  to  find  the  less. 

Rule.  Subtract  one  from  the  other. 

If  the  greater  number  f)e  672,  and  the  difl^erence  between 
that  and  the  less  389,  required  the  other  number,  672  given— 
389  diflrerence=283  less. 

3.  Having  the  least  of  two  numbers  given,  and  the  diff*erence 
between  that  and  the  greater,  to  find  the  greater. 

Rule. — Add  them  together. 

The  less  number  is  142,  and  the  difference  167;  required 
.he  greater  number.     309  greater  number. 


APPLICATION    OF    THE    COMPOUND    RULES.  87 

4.  Having  the  product  of  two  numbers,  and  one  of  them 
given,  to  find  the  other. 

Rule.  Divide  the  product  by  the  given  number,  and  the  quo- 
tient v^rill  be  the  number  required. 

If  the  product  of  two  numbers  be  196,  and  one  of  them  4,  re- 
quired the  other.     Thus:   196-4-4=49.     Ans. 

5.  Having  the  dividend  and  quotient,  to  find  the  divisor. 

Rule.  Divide  the  dividend  by  the  quotient.  This  will  prove 
division. 

Let  the  dividend  be  144,  and  the  quotient  16,  required  the 
divisor?     144—16=9.     Ans. 

6.  Having  the  divisor  and  quotient  given,  to  find  the  dividend. 

Rule.  Multiply  them  together. 

Let  the  divisor  be  6,  and  the  quotient  72,  required  the  divi- 
dend.    72X6=432.     Ans. 

EXAMPLES. 

1.  Suppose  a  man  born  in  the  year  1743,  when  will  he  be 
77  years  of  age  ?  Ans.  1820. 

2.  What  number  is  that,  which  if  it  be  added  to  19418  will 
make  21802?  Ans.  2384. 

3.  What  number  must  you  multiply  by  9,  that  the  product 
may  be  675  ?  Ans.  75. 

4.  What  is  the  difference  between  thrice  five  and  thirty,  and 
thrice  thirty-five  ?  ^  Ans.  60. 

5.  If  a  man  spend  192D.  in  a  year,  how  much  is  that  per  cal- 
endar month  ? 

6.  A.  borrowed  at  different  times  the  following  sums,  namely : 
of  B.  D625  ;  of  C.  D721.50  ;  of  D.  D842  ;  and  he  is  indebted 
to  others  as  much  as  he  has  borrowed,  abating  D125.50  ;  he  is 
now  prepared  to  retire  from  business  ;  required  the  amount  of 
his  debts.  Ans.  D4251.50 

7.  General  Burgoyne  and  his  army  were  captured  at  Saratoga, 
N.  Y.,  by  General  Gates,  October  17,  1777,%nd  Earl  Cornwallis 
and  his  army  surrendered  to  General  Washington  at  York,  Va., 
October  19,  1781  ;  required  the  space  of  time  between. 

Ans.  4  years,  2  days 

8.  Bought  5  loads  of  wood :  the  first  containing  1  cord,  32 
feet ;  the  second,  1  cord,  64  feet ;  the  third,  112  feet ;  the  fourth, 
1  cord,  28  feet ;  the  fifth,  1  cord,  20  feet ;  how  much  in  all  ? 

Ans.  6  cords 


88  REDUCTION    OF    DECIMALS. 

9  How  many  bottles  holding  1  pint,  2  gills  each,  are  required 
for  bottling  4  barrels  of  cider  ? 

10.  9  A.  7  R.  50po.+  17  A.  IIR.  70po.;  then-12A.  5R. 
45  po. ;  then  X  by  62  ;  then  -^-by  49. 


REDUCTION  OF  DECIMALS. 

The  two  following  cases  of  Reduction  of  Decimals  are  the 
very  reverse  of  each  other,  and  by  reversing  the  rule,  one  will 
prove  the  other.  The  use  of  the  rule  is  to  change  a  denomina- 
tion, or  several  denominations,  from  their  given  expression  into 
a  decimal  quantity  having  the  same  value,  for  the  purposes  of 
multiplication,  division,  &c.,  which  will  then  become  the  same 
as  whole  numbers.  Thus,  if  you  wish  to  find  the  decimal  ex- 
pression of  2  quarters,  14  pounds,  avoirdupois,  first  reduce  it  to 
pounds,  2  x28=:56+  14=70  pounds  ;  now  annex  ciphers  to  the 
70  pounds,  and  divide  by  112  lbs.=z=l  cwt.  of  which  you  wish 
to  make  it  a  decimal,  and  you  have  .625;  jiwo—TT2~i^^^ 
JDy^Y^^f.  Again,  you  wish  to  reverse  the  rule,  and  give  the 
.625  the  expression  in  the  terms  of  the  integer,  or  to  reduce  it 
to  its  proper  value.  Thus  .625  cwt.  multiply  by  the  next  less 
denomination,  which  is  4  qrs.,  and  count  off  the  three  places  for 
decimals,  according  to  the  rule  in'  multiplication  of  decimals, 
and  the  quantity  at  the  left  of  the  decimal  point  is  of  the  same 
name  with  the  multiplier,  that  is,  2  qrs.  ;  then  multiply  the  re- 
maining decimal,  .500,  by  the  next  less  denomination,  namely, 
28  lbs.,  and  count  off  as  before,  and  you  have  2  quarters,  14 
pounds  =  j%2_5__^7^o__  5  c^i-^  ^c.  The  correctness  of  the  op- 
eration of  the  rule  is  evident  from  the  nature  of  decimals,  and 
the  explanations  already  given  in  the  preceding  pages. 

In  the  third  case  :  to  reduce  a  vulgar  fraction  to  its  equiva- 
lent decimal,  by  annexing  one,  two,  three,  or  more  ciphers  to 
the  numerator,  the  value  of  the  fraction  is  increased  ten,  a  hun- 
dred, or  more  times.  After  dividing,  the  quotient  will,  of  course, 
be  ten,  a  hundred,  or  more  times,  too -much  ;  the  quotient  must, 
therefore,  be  divided  by  ten,  a  hundred,  (fcc,  to  give  the  true 
quotient  or  fraction.  In  the  first  example,  ^  is  i<^^  =  i^*, 
which  divided  by  1000  is  ^^^j^z=:.l25,  and  this  is  the  rule. 


REDUCTION    OF    DECIMALS.  89 

To  reduce    numbers  of  different  denominations,  as  of  money, 
weight,  measure,  <Sfc.,  to  their  equivalent  decimal  value. 

RULE. 

1 .  Begin  at  the  left  hand  and  multiply  by  as  many  as  it  takes 
of  the  next  lower  denomination  to  make  one  of  the  higher,  add- 
ing in  the  denominations  respectively,  as  you  multiply,  until 
they  are  reduced  to  the  lowest  denomination  in  the  question, 
and  this  is  the  dividend. 

2.  Then  take  one  of  that  denomination  of  which  you  wish  to 
make  it  a  decimal,  and  reduce  it  to  the  same  denomination  with 
the  one  above-mentioned,  and  this  last  number  is  the  divisor. 
Divide  as  in  whole  numbers,  and  the  quotient  is  the  answer 
Put  the  point  before  it,  for  it  is  always  a  decimal. 

EXAMPLES. 

1.  Reduce  3  roods,  20  poles,  to  the  decimal  of  an  acre. 

lA.  Explanation. — 

4R.  X   R.     po.  .875 A.  You  will  first  mul- 

77         3       20  4R.=1A.     tiply  the    3  roods 

^^       40  X  by  40,  because  40 

R.  3.500  poles  make  1  rood, 

po.  1 60)  140.000(-Yyjy\  Ans.         40po.=lR.  and  add  in  the  20 

1280  po.,  this  will  make 

po.  20.000  proof.       140    poles ;     then 

1200  annex  ciphers,  and 

1120  the    dividend   will 

be    formed ;     now 

800  *  you  wish  to  make 

800  it  the  decimal  of  an 

an  acre,  then  take 

1  acre  and  reduce  it  to  the  same  denomination  as  the  dividend, 
that  is,  poles,  160  =  1  acre,  and  this  is  the  divisor ;  then  to  prove 
it,  multiply  the  quotient  by  the  next  lower  denomination,  observ- 
ing to  count  off  for  the  decimal  point,  and  the  denominations  at 
the  left  of  the  point  make  the  answer,  &c. 

2.  Reduce  20  poles  to  the  decimal  of  an  acre.  Ans.  .125. 
Reduce  2  R.  4  po.  to  the  decimal  of  an  acre.  Ans.  .525. 
Reduce  3  quarters,  2  nails,  to  the  decimal  of  a  yard. 

4  qrs.  3  qrs.  2  na. 

4  na.  X  4 

nails,  16  divisor.  16)14.000(.875  Am. 

5.  Reduce  1  gall,  to  the  decimal  of  a  hhd.        Ans.  .015873 

8* 


90  REDUCTION    OF    DECIMALS. 

6.  Reduce  7  oz.  19  dwt.  to  the  decimal  of  a  pound  troy. 

240)159.0000(.6625.  Ans 

7.  Reduce  3  quarters,  21  pounds,  avoirdupois,  to  the  decimal 
of  a  cwt.  Ans.  .9375, 

8.  Reduce  2  feet,  6  inches,  to  the  decimal  of  a  yard  ? 

Ans,  .83333+ 

9.  Reduce  5  furlongs,  16  poles  to  the  decimal  of  a  mile. 

Ans.  .675 

10.  Reduce  4.5  calendar  months  to  the  decimal  of  a  year. 

Ans.  .375. 

11.  Reduce  40  minutes  to  the  decimal  of  an  hour.        .666 -j- 

12.  Reduce  20  minutes  to  the  decimal  of  an  hour.        .333-f- 

13.  Reduce  1  day,  6  hours,  to  the  decimal  of  a  week.       .1785. 

14.  Required  the  cost  of  1  yard,  2  quarters,  2  nails  of  cloth, 
at  D5.00  per  yard. 

2  qrs.  2  na.  yd.  1.625 

4  5 


16)10.000(.625.  D8.12.5  Ans. 

15.  What  is  the  value  of  3  yards,  1  quarter,  3  nails  of  cloth, 
al  D4.50  per  yard? 

1  qr.  3  na.  yd,  3.4375 

4  4.50D. 


16)7.0000(.4375.  D15.46.87.5  Ans. 

16.  Required  the  value  of  1  acre,  2  roods,  18  poles,  at  D20 
per  acre  ? 

2  roods  18  poles.  acres  1.6125 

40  20D. 


poles  160)98.0000(.6125  D32.25.00  Ans. 

17.  What  cost  2  acres,  1  rood,  30  poles,  at  D25  per  acre  ? 

Ans.  D60.93.75 

18.  What  cost  2  pecks,  7  quarts  of  wheat,  at  Dl.50  per  bush 

Ans.  Dl. 07.8.125. 

19.  What  cost  7  bushels,  2  pecks,  7  quarts  of  corn  at  52  cts. 
per  bushel?  D4.01.375. 

20.  What  is  the  cost  of  3  bushels,  3  pecks,  4  quarts  of  rye, 
at  75  cents  per  bushel?  Ans.  D2.90.6.25. 

21.  What  is  the  cost  of  2  quarters,  15  pounds  of  coffee,  at 
Dll  per  cwt.  ?  Ans.  D6.97.312. 

22.  Reduce  375678  feet  to  miles   and  decimals.      71.151-f 
Note. — This  last  question  is  an  exception  to  the  general  rule, 

©r  at  least  questions  of  this  kind  are  not  of  frequent  occurrence 


REDUCTION    OF    DECIMALS.  91 

although  the  rule  will  apply  in  this  as  well  as  in  other  cases ; 
the  number  of  feet  given  are  equal  to  a  number  of  miles  ;  there- 
fore, take  the  feet  for  the  dividend,  reduce  1  mile  to  feet=5280 
for  a  divisor,  then  divide  according  to  the  rules  of  Division  of 
Decimals,  and  you  have  71.151-1-  miles.  This  example  will 
apply  and  be  sufficient  in  cases  of  this  kind. 

To  reduce  a  Decimal  to  its  proper  value. 

RULE    IT. 

1.  Multiply  the  decimal  by  the  number  of  parts  in  the  next 
less  denomination,  and  cut  off  so  many  places  for  a  remainder, 
counting  from  the  right,  as  there  are  decimal  places  in  the  given 
decimal,  and  there  make  the  decimal  point. 

2.  Multiply  the  remainder,  that  is,  the  decimal,  by  the  next 
less  denomination,  and  cut  off  a  remainder  as  before  ;  continue 
in  this  way  through  all  the  parts  of  the  integer,  and  the  several 
denominations  standing  on  the  left  hand  of  the  decimal  point 
make  the  answer. 

23.  Required  the  value  of  .9075  of  an  acre. 

Explanation. — Begin  and  multiply 
by  the  next  lower  denomination  that, 
the  one  mentioned  in  the  question 
which  is  roods,  4  of  which  make  ap 
acre.  After  multiplying,  count  off  a* 
many  places  or  figures  as  there  are 
in  the  given  question  for  decimals,  and 
there  place  the  decimal  point,  and  the 
figure  or  figures  at  the  left  of  the 
decimal  point,  will  be  of  the  same 
name  as  the  multiplier,  namely,  roods. 
Then  take  the  next  less  denomination, 
which  is  poles,  40  of  which  make  1 
rood ;  multiply  all  the  figures  on  the 

right  of  the  decimal  point,  then  count 

1800000  off  from  right  to  left  as  many  figures 

1800000  as  there  were  decimals  in  the  number 

450000  multiplied ;    there  place  the  decimal 

In.     64.800000  point,  and  the  figures  at  the  left  of  the 

point  will  be  poles.    Thus  far  we  have 

3  roods,  25  poles,  and  .2000 ;  now  30.25  yards  make  one  pole 

in  square  measure  ;  therefore,  multiply  by  that  number  those 

figures  at  the  right  of  the  decimal  point,  which  are  .2000,  and 

count  off  in  the  product  4  figures  for  those  in  the  multiplicand, 

and  2  for  the  multiplier,  6  figures,  and  there  place  the  decimal 


.9075 
X4R. 

Roods 

3.6300 
40  po. 

Poles,    25.2000 

30.25  yd. 

Yards,  6.050000 

9  feet 

Feet, 

0.450000 

144  in. 

92  REDUCTION    OF    DECIMALS. 

point,  and  you  have  6  yards  ;  then  multiply  by  9,  because  9 
feet  make  1  yard,  and  count  off  as  before,  which  gives  0  feet^ 
yet  the  remaining  decimal  of  a  foot  must  be  reduced  to  inches, 
144  being  equal  to  1  foot,  and  you  have  64  inches.  Thus  we 
find  that  .9075  of  an  acre  is  equal  to  3  roods,  25  poles,  6  yards, 
0  feet,  64  inches. 

24.  What  is  the  value  of  .046875  of  a  pound  avoirdupois  ? 
Thus,  .046875x16=750000x16—12.000000  drams,  Ans. 

25.  What  is  the  value  of  .617  of  a  cwt.  ? 

Ans.  2qr.  131b.  1  oz.  10  dr. 

26.  What  is  the  value  of  .569  of  a  year  of  365  days  ? 

Ans.  207  da.  16  ho.  26  min.  24  sec. 

27.  Find  the  proper  quantity  of  .9071  of  an  acre. 

28.  Find  the  value  of  .76442  of  a  pound  troy. 

Ans.  9  oz.  3  dwt.  11  gr. 

29.  What  is  the  value  of  .875  of  a  yard  ?        Ans.  3  qr.  2  na. 

30.  Find  the  proper  quantity  of  .089  of  a  mile  1 

Ans.  28  po.  2  yds.  1  ft.  11  in.+ 

31.  Find  the  proper  quantity  of  .897  of  a  degree. 

To  reduce  a  Vulgar  Fraction  to  its  equivalent  decimal, 

RULE    III. 

1 .  Annex  ciphers  to  the  numerator,  and  then  divide  it  by  the 
denominator. 

2.  There  must  be  as  many  places  pointed  off  in  the  quotient 
as  you  annex  ciphers  to  the  given  numerator ;  if  there  should 
not  be  so  many  places  of  figures  in  the  quotient,  the  deficiency 
must  be  supplied  by  prefixing  ciphers  at  the  left  hand  of  the 
quotient ;  if  the  numerator  is  larger  than  the  denominator,  there 
must  be  a  whole  number,  or  whole  number  and  decimal. 

32.  Reduce  i  to  a  decimal  having  the  same  value  1       8)1.000 

Ans.  .125 

33.  What  decimal  is  equal  to  J?  2)1.0 

^  — 

Ans.  .5 

34.  Reduce  ^i  to  a  decimal.  Ans,   .6875. 

35.  Reduce  i  to  a  decimal.  .2. 

36.  Reduce  ^j  to  a  decimal.  .037037  +  . 

37.  Reduce  ^  to  a  decimal.  .33333  +  . 

38.  Reduce  -^^  to  a  decimal.  .85. 

39.  Reduce  3%  to  a  decimal.  .09375. 

40.  Reduce  jyVs  ^^  ^  decimal.  .008 

41.  Reduce  /^  to  a  decimal.  .1923076 


REDUCTION    OF    DECIMALS.  93 

42.  Reduce  ^^  to  a  decimal.  Ans,  .153846 

43.  Reduce  }|  to  a  decimal.  -25=:^ 

44.  Reduce  ^^  to  a  decimal.  .125  =  1 

45.  Reduce  ^-|  to  a  decimal.  .1875. 

46.  Reduce  /g-  to  a  decimal.  .083333  + 

47.  Reduce  ^§  to  a  decimal. 

48.  Reduce  %*  to  a  whole  number.  8.00 

49.  Reduce  ||-  to  a  whole  number. 

50.  Reduce  y/  to  a  whole  number. 

'  51.  Reduce  ^U®  to  a  whole  number. 

REVIEW    OF    DECIMALS.    NO.    11. 

.  When  you  wish  to  reduce  several  denominations  to  the  deci- 
mal of  a  larger  denomination,  still  retaining  the  same  value,  how 
do  you  proceed?  (See  rule,  (fee.)  In  question  1st  the  quotient  is 
.875  ;  what  do  you  understand  by  this  ?  Ans.  Eight  hundred 
seventy-five  thousandths  of  an  acre,  which  is  equal  to  140  rods  • 
that  is,  140  rods  bear  the  same  proportion  to  160  rods,  or  one 
acre,  as.  875  does  to  .1000,  or  y^^^^,  or  875  to  1000,  &c.  Sup- 
pose the  number  at  the  left  of  the  decimal  point  in  the  dividend 
had  been  more  than  160  (the  divisor)  would  the  quotient  have 
been  a  decimal  ?  Ans.  In  that  case  it  must  have  been  a  whole 
number  and  decimal,  but  after  a  decimal  figure  is  brought  down 
for  a  new  dividend  that  quotient  figure  will  be  a  decimal,  as  there 
can  be  no  decimal  in  the  quotient  until  there  is  one  in  the  divi- 
dend. How  do  you  reduce  a  decimal  to  its  proper  value  1  (See 
the  rule.)  Can  you  explain  the  first  example  ?  Wherein  does 
this  rule  differ  from  the  one  immediately  preceding  it  ?  Ans. 
It  is  the  very  reverse  of  the  first,  and  they  reciprocally  prove 
each  other.  How  do  you  reduce  a  vulgar  fraction  to  it*?  equiva- 
lent decimal?  What  is  the  rule  ?  What  do  you  understand  by 
numerator?  Ans.  It  is  the  number  above  the  line,  thus  ^, 
which  serves  as  the  common  measure  to  others.  What  is  the 
meaning  of  denominator  ?  Ans.  It  is  the  number  below  the 
line,  thus  ■§-,  and  is  the  number  of  parts  a  unit,  or  anything,  is 
divided  into,  and  gives  the  name  or  denomination  of  the  parts. 
What  i^  understood  by  the  expression  of  |-D.  ?  ^^^qD.  ?  Ans. 
Seven  eighths  of  a  dollar,  or  87.5  cents,  and  the  other  is  twenty- 
five  hundredths,  which  is  25  cents.  If  you  owned  If  of  the  cap- 
ital in  trade,  what  part  would  be  yours  ?  Ans.  Two  thirds  (§•); 
that  is,  the  stock  would  be  divided  into  24  shares,  and  I  should 
have  sixteen  of  them,  and  16  is  two  thirds  of  24.  Suppose  the 
stock  divided  into  32  shares,  what  part  would  then  be  yours  ? 
Ans.  One  half,  because  \^  is  equal  to  one  half.     If  divided  into 


94  APPLICATION    OF    THE    RULES    OF    DECIMALS. 

64  shares  how  much  1     Ans.  One  fourth  (J)  ;  for  as  the  de 
nominator  increases,  the  vahie  of  the  numerator  decreases  in  the 
same  proportion  ;  and  the  same  of  the  denominator,  for  ^|-,  in 
stead  of  being  one  half  as  above,  it  would  be  just  4  times  as 
much=D2.00,  or   my  share   would  be  4xl6=:64,  which  is 
more   than  the  given  capital.     What  is  this  last  (^|-)  fraction 
called?     Ans.  An  improper  fraction.     Why?     Ans.  Because 
the  numerator  is  larger  than  the  denominator,  consequently  the 
fraction  expresses  one  whole,  or  more  than  one  whole  number, 
as  f  =1,  V^  =  2,  &c.     Any  number  may  be  expressed  or  writ-, 
ten  fractionwise,  wheiher  it  be  a  whole  number,  decimal,  or  vul- 
gar fraction,  retaining  its  value,  thus:  J§tH-50=-3|Q  ;  fl-T-S 
=/^rr.06974+  ;  f§~8=f=1.25;  746-^f  =1989i,  &c. 

Note.  This  part  of  arithmetic  is  fully  explained  in  its  proper 
place.     (See  Vulgar  Fractions.) 


APPLICATION  OF  THE  RULES  OF  DECIMALS,  &c, 

1.  Add  163-7-  days,  183-2^%,  143-7^%  together. 

2.  A.  purchased  4  hogsheads  of  molasses  :  the  first  contained 
62y\^  gallons  ;  second,  72^q^-^-^q  gallons  ;  third,  50j2_  gallons  ; 
fourth,  55-^-^Q  gallons  ;  how  many  gallons  in  all  ? 

Ans,  240.6157. 

3.  What  is  the  amount  of  one  and  five  tenths  ;  forty-five  and 
three  hundred  and  forty-nine  thousandths ;  and  sixteen  hun- 
dredths ?  Ans,  47.009. 

4.  What  is  the  difference  between  4.75  minutes  and  6.25 
minutes?  Ans.  1.5  mi. 

5.  From  .65  cwt.  take  .125  cwt.  Ans.  .525. 

6.  From  769.7  hogsheads  take  596.92.  Ans.  172.78. 

7.  What  will  9.7  lb.  of  sugar  cost  at  D0.07  per  lb.  ? 

Ans.  DO.67.9. 

8.  At  D0.90  per  bushel,  what  will  7.25  bushels  cost? 

Ans.  D6.52.5. 

9.  What  is  the  quotient  of  1561.275-^-24.3  ?        Ans.  64.25 

10.  What  is  the  quotient  of  .264-^2?  Ans.. 132. 

11.  What  decimal  is  equal  to  f  ?  Ans.  .25 

12.  What  decimal  is  equal  to  -^q  ?  .175. 


APPLICATION    OF    THE    RULES    OF    DECIMALS.  95 

13.  What  decimal  is  equal  to  ^  ?  Ans,  .3333,  circulating  rept. 

14.  What  is  the  value  of  2  yds.  3  na.  French  calico,  at  70  c. 
per  yd  ''  Ans.  Dl.53.125. 

15.  What  cost  2  cwt.  17  lb.  of  chocolate,  at  15D.  per  cwt.  ? 

2.151786  X15  =  D32.27.6790.  Ans. 

16.  What  cost  1.25  tons  of  hay,  at  llD.  per  ton  ? 

Ans.  13D.  75c. 

17.  If  31.25  yds.  cost  75D.  how  much  is  it  per  yard  ? 

Ans.  D2.40. 

18.  If  35  bushels  of  wheat  cost  D97.50,  what  did  1  bushel 
cost  ?  Ans.  D2.78.5  + 

19.  What  is  the  cost  of  1  gall,  and  1  pt.  of  wine  at  D1.75 
per  gall.?  Ans.  D  1.96.875. 

20.  If  I  pay  14D.  for  112  lb.  of  sugar,  and  sell  it  for  12.5c. 
per  pound,  do  I  gain  or  lose,  and  how  much  ? 

21.  If  I  pay  7D.  for  80  pounds  of  flax,  »nd  sell  it  for  8c.  per 
pound,  do  I  gain  or  lose,  and  how  much  ?  Ans.  Loss  60c. 

22.  When  rye  is  selling  at  Dl.25  per  bushel,  how  many 
bushels  can  I  have  for  D62.50.  Ans.  50  bushels. 

23.  A  gentleman  bequeathed  his  estate  of  one  hundred  thou- 
sand dollars  to  his  son  and  daughter  ;  the  son  was  to  have  15 
thousand,  15  hundred  and  15  dollars  more  than  the  daughter; 
required  the  share  of  the  daughter.  Ans.  41742.50. 

24.  If  the  sum  of  33485D.  was  expended  in  the  purchase  of 
United  States  land,  at  Dl.25  per  acre,  how  many  acres  would 
it  purchase  1  Ans.  26788  acres. 

25.  Then  if  you  should  sell  the  26788  acres  for  D2.5  per 
acre,  how  much  would  you  gain  ?  Ans.  33485D. 

26.  If  you  should  buy  a  chest  of  tea  containing  135  lb.  for 
150D.,  and  sell  the  same  for  Dl.30  per  lb.,  what  would  be  the 
difference  ?  Ans.  Gain  D25.5. 

27.  If  a  pipe  of  wine  contains  126  galls.,  how  many  pinta 
are  in  the  pipe,  and  what  is  it  worth  at  14.5c.  per  pt.  ? 

Ans.  1008  pints,  worth  D146.16. 

28.  K  farmer  purchased  120  sheep,  for  which  he  paid  D1.75 
per  head,  and  47  young  cattle  at  D  14.50  per  head  ;  he  sold  the 
sheep  for  D2.14  per  head,  and  the  cattle  for  14D.  per  head  ;  did 
he  gain  or  lose,  and  how  much  ^  Ans.  Gained  D23.30. 

29.  A  certain  field  will  produce  127  bushels  of  grain;  how 
much  will  a  field  produce,  just  seven  and  a  half  times  as  large  ? 

Ans.  952.5  b. 

30.  A.  and  B.  start  from  the  same  place  ;  A.  goes  east  17  days, 
at  the  rate  of  39  miles  per  day ;  B.  goes  west  at  the  rate  of  43 
miles  per  day  for  14  days  ;  how  many  miles  will  each  travel, 


96  APPLICATION    OF    THE    RULES    OF    DECIMALS. 

and  how  many  distant  from  each  other  ?  A71S.  A.  663  m.,  B.  602 
m.,  distance,  1265  m. 

31.  If  57  yards  of  cloth  cost  197D.,  what  cost  1  yard? 

32.  If  18  yds.  1  qr.  of  cloth  cost  D91.16,  what  is  the  cost 
of  1  yd.  ? 

33.  If  you  have  7D.  and  pay  away  .07,  how  much  will  re 
main  ? 

34.  What  is  the  value  of  23  grains  of  silver,  at  14D.  per  lb 
troy  ? 

35.  If  you  earn  ID.  and  Im.  per  day,  how  much  is  that  h 
one  year  1 

36.  John  Armstrongs  Dr. 

To  1  pound  of  tea  at  D  1.56c. 

To  10  pounds  loaf  sugar  at  19c. 

To  3  gallons  of  wine  at  Dl.17. 

To  9  pounds  of  steel  at  10c. 

To  6i  yards  of  sheeting  at  54c. 

To  3  yards  of  cotton  at  18c. 

To  2  pounds  of  butter  at  31c. 

To  1  pound  of  indigo  at  Dl.46. 

To  goods  per  order,  D3.72. 

To  4  bushels  of  oats  at  50c.  D19.72. 


37.  Levi  Woodbury^  Cr. 

By  5.5  bushels  rye  at  50c,  -  D2.75 

By  3i  cords  of  wood  at  1)4.50  -  15.75 

By  3.25  bushels  of  wheat  at  Dl.75  -  5.68.7.5 

By  cash  -  2.75 

By  25  pounds  of  cheese  at  7c.  -  1.75 

By  20.5  pounds  of  butter  at  18c,  -  3.69 

By  li  cwt.  of  iron  at  D6.58  -  8.22.5 

By  12  pounds  of  lard  at  12ic.  -  1.50 

By  7^  pounds  of  raisins  at  9c.  -  67.5 

By  6  quarts  of  salt  at  5c.  -  30 


38.  How  many  quarts  in  7  hogsheads  ?  in  18  ?  in  27  ?  in  34  ? 

39.  How  much  is  a  hogshead  of  molasses  worth,  at  9c.  per 
quart?  at  14? 

40.  A  cask  of  sugar  weighs  5  cwt.,  what  is  it  worth  at  12^c. 
per  pound  ?  Ans.  D70.00. 

41.  If  .25  of  a  yard  is  worth  D0.25,  what  cost  17.5  yards  ? 
what  cost  27.25  yards  ? 

42.  How  many  cubic  inches  in  5  feet?  in  11  feet  ?  in  40  ft.  ? 

43.  How  many  inches  in  35  poles  ?  in  65  poles  ?  in  75  poles  ? 


Kj-:dUCTIO.S     of    DIFFERENT    DENOMINATIONS.  0'^ 

44.  How  many  secpnds  in  5  hours  ?  in  2  days  ?  in  1  week  ? 

45.  How  many  hours  in  5  years?    in  12  years?    in    170 
days  ? 

46.  How  many  rods  in  7  acres,  and  how  much  would  it  be 
worth  at  75  cents  per  rod  ?  Arts,  840 D. 

47.  In  6795  square  feet,  how  many  square  yards  ? 

Ans.  755. 

48.  In  78479  mills  how  many  dollars  ?  how  many  eagles  ? 

49.  What  cost  60  galls.  1  pt.  of  wine,  at  .75D.  per  gallon  ? 

Ans.  D45.09.34- 

50.  What  will  26  yards  2  quarters  of  cloth  cost  at  D4.75  per 
yard?  Ans.  D125.87.5. 

51.  Required  the  value  of  25.874  yards;  39.42  yards ;  67 
yards,  2  nails  ;  47  yards,  2  quarters,  at  D3.074  per  yard. 

52.  What  cost  91.27  gallons;  218.087  gallons;  8  gallons, 
2  quarts  ;  91  gallons,  3  quarts,  at  Dl.25  per  gallon  ? 

53.  47963.0521 +897.009  +  81.81 +9.0009  +  .629108-M.08. 

54.  What  number  is  that,  which  being  added  to  9.0146  will 
make  20.7092  ? 


REDUCTION  OF  DIFFERENT  DENOMINATIONS. 

In  Reduction  of  Different  Denon^inations  we  have  several 
denominations  given  to  be  reduced  to  a  lower  denomination, 
still  retaining  the  same  value  ;  it  is  called  compound  reduction. 
The  remarks  and  explanations  in  simple  reduction  will  apply  ou 
the  present  occasion. 


1 .  Multiply  the  highest  denomination  by  as  many  of  the  next 
hess  as  it  requires  to  make  one  of  that,  and  add  in  the  second 

denomination,  if  any  ;  then  multiply  by  the  next  less  denomina- 
tion, and  add  in,  if  any  ;  continue  in  this  way  through  the  sum. 

2.  When  small  denominations  are  to  be  reduced  to  larger  de- 
nominations, reverse  this  operation,  and  begin  by  dividing  in  the 
same  manner  as  you  are  directed  above  to  multiply ;  by  this 
reversion  of  the  operatior.  the  proof  is  obtained.  See  the  ev 
amples. 

9 


98 


REDUCTION    OF    DIFFERENT    DENOMINATION^. 


EXAMPLES    AND  QUESTIONS. 

4  28  16         16 

1.  Reduce  5  cwt.  2  qrs.  12  lb.  11  oz.  10  drs.  to  drams. 
4x 
—  16)1609^1^  drams. 

20  

2+  16)10059+10  dr. 


22  qrs 
28  X 

176 

28)628+11  oz. 
4)22  +  12  lb. 

44 

12  + 

628  lb. 

cwt.  5+2  qr.  proof 
5  cwt.  2  qrs.  12  lb.  11  oz.  10  dr. 

16x 

Explanation.     Multiply  the  5  cwt. 

3768 
628 

10048 
11  + 


10059  oz. 
16x 


60354 
10059 
10  + 


by  4  quarters,  and  add  in  the  2  quarters 
and  you  have  22  quarters  ;  then  multi- 
ply 22  quarters  by  28  pounds,  and  add 
in  the  12  pounds,  and  you  have  628 
pounds  ;  then  multiply  628  pounds  by 
16  ounces  and  add  in  the  11  ounces, 
and  you  have  10059  oz.  ;  then  multi- 
ply those  ounces  by  16  drams,  and  add 
in  10  drams,  and  you  have  160954, 
which  is  the  answer ;  then  for  the 
proof  you  can  divide  by  short  or  long 
division,  as  you  like,  &c. 


160954  drams. 
*Z    In  126230400  seconds  how  many  years  of  365  days  ? 
6,0)12623040,0 


6,0)210384,0 


4X6=24  h. 


4)35064 


I    6)8766 

365)1461(4  years. 
1460 


Ans,  4  years,  1  day. 


}  4ay. 


REDUCTION    OF    DIFFERENT    DENOMINATIONS.  % 

3.  Reduce  3124446  drams  to  tons 
{  4)3124446 


4X4=16  drams  < 


4x4  =  16  oz. 


4X7=28  lb. 


(    4)781111  ...  2 

(     4)195277  ...  3x4+2  =  14  dr. 

(      4)48819  ...  1 


4)12204  .  .  .  3x4+1  =  13  oz. 


7)3051 


qrs.  4)435  ...  6x4=24  lbs 

ewt.  2,0)10,8  ...  3  qrs. 

5  T.  8  cwt. 
Ans.  5  T.  8  cwt.  3  qrs.  24  lbs.  13  oz.  14  drs 

4  Reduce  2  cwt.  3  qrs.  17  lbs.  to  pounds.         Ans.  325  lbs. 

5  Bring  16  bushels  1  peck  to  pecks.  65  pks. 

6  Reduce  2  hhds.  10  gallons  to  quarts.  544  qts. 

7.  Reduce  18  years,  6  months,  to  months.  222  mo. 

8.  Reduce  15  yards,  2  feet,  to  inches.  564  in. 

9.  Reduce  3  leagues,  2  miles,  7  furlongs,  to  furlongs    95  fur. 

10.  Reduce  11  acres,  2  roods,  19  poles,  to  poles.         1859  po. 

11.  In  12  cords  of  wood  how  many  solid  feet  and  inches  ? 

Ans.  1536  ft.;  2654208  in. 

12.  In  21  cords  of  wood  how  many  solid  feet  ?  2688  ft, 

13.  In  24796800  seconds,  how  many  weeks  ?  41  w. 

14.  In  20692  square  rods,  how  many  acres  ?   129  A.  1  R.  12  po. 

15.  In  15840  yards  how  many  miles  and  leagues  ?      9  m.  3  1. 

16.  In  51  miles  how  many  furlongs  and  poles  ? 

Ans.  408  fur.;  16320  po 
J7.  In  3783  nails  how  many  yards  ?  236  yds.  1  qr.  3  na. 

18.  Bring  892245  ounces  into  tons. 

Ans.  24  T.  17  cwt.  3  qr.  17  lb.  5  oz. 

19.  In  12  hhds.  of  sugar,  each  11  cwt.  25  lb.,  how  many 
ounces  ?  Ans.  241344  oz. 

20.  How  many  revolutions  will  the  wheel  of  a  steamboat,  27 
feet  in  circumference  make,  in  navigating  the  Hudson  from 
New  York  to  Albany,  supposing  the  distance  160  miles,  with- 
out making  any  allowance  for  current  or  tide  ?  Ans.  31288. 


100  REDUCTION'    OF    DIFFERENT    DENOMINATIONS. 

21.  How  many  inches  between  the  earth  and  the  sun,  the 
distance  being  95,000,000  miles  ?  Ans.  6019200000000. 

22.  If  a  cannon  ball  move  at  the  rate  of  1  mile  in  7  seconds, 
how  long  wonld  it  be  in  passing  from  the  earth  to  the  sun  1 
(Multiplyby  7,divideby60,&;c.),l;i?i5.  21yr.31da.  18h.13m.20s. 

23.  Admitting  your  age  to  be  16  years,  1  week,  17  days,  20 
minutes,  45  seconds,  how  many  seconds  old  are  you  ?  (52w.=  ly.) 

Ans.  50526S445S. 

24.  Reduce  184320  grains  to  pounds.  Ans.  32  lbs. 

25.  In  J  6  tons  how  many  hundred  w^eight  ?  quarters  ?  pounds  ? 
ounces  1  and  drams  ? 

A;^^.  320cwt. ;  1280qrs.;  320001bs.;  51200007..;  8192000drs. 

26.  In  1332005  grains  (apoth.)  how  many  pounds  ? 

Ans,  231  lb.  3  oz.  5  gr. 

27.  In  9173  nails  how  many  yards  ?    Ans.  573  yds.  1  qr.  1  na. 

28.  In  3  ells  French,  how  many  inches  ?  A7is.  162. 

29.  in  34594560  inches  how  many  miles  ?  Ans.  546  ra. 

30.  What  is  the  circumference  of  the  earth  in  yards  ? 

Ans.  44035200  yds. 

31.  Reduce  a  solar  year  to  seconds.         Ans.  31556937  sec. 

32.  In  622080  solid  inches  how  many  tons  of  round  timber? 

Ans.  9  T. 

33.  In  5529600  cubical  inches  how  many  cords.       Ans.  25. 

34.  How  many  minutes  and  seconds  in  one  complete  revolu- 
tion of  a  planet  ?  Ans.  21600  minutes  ;  and  1296000  sec. 

35.  In  89763  square  yards  how  many  acres  ? 

A?is.  18  A.  2  R.  7  po.  101  ft.  36  in. 

36.  In  5054  pints  how  many  bushels  ?  A?is.  78  b.  3  pk.  7  qts. 

37.  How  many  quarters  of  rice,  at  6c.  per  pound,  may  be 
bought  for  D3.36  ?  Ans.  2  qrs. 

38.  How  many  tuns  of  wine,  at  6  cents  a  gill,  may  be  bought 
for  D483.84.  Ans.  1  tun. 

39.  What  is  the  value  of  a  silver  cup  weighing  10  oz.  5  pw^t. 
18  gr.  at  5  mills  per  grain  1  Ans.  24D.  69c. 

40.  At  9  cents  an  hour  what  will  2  yrs.  6  mo.  3  w.  6  da.  12 
h.  labor  be?  Ans.  D  1873.80. 

41.  At  320  cents  a  yard,  what  will  64  nails  of  cloth  cost? 

Ans.  D  12.80. 

42.  In  86400  grains  how  many  scruples  ?  drams  ?  and  ounces  ? 

^71^.  4320  sc. ;   1440  dr.  ;   180  oz 

43.  In  8903304  barleycorns  how  many  leagues  ? 

Ans.  15  1.  1  m.  6  fur.  28  po.  4  yds. 

44.  In  20  cords  of  wood  how  many  solid  inches  ? 

Ans.  4423680  in. 


PROPORTION  ;    OR,    SINGLE    RULK    OF    THREE.  101 

io.  In  20  tuns  of  wine  how  many  gills  J         Ans.  161280  g 

46.  In  11378955037  seconds,  how  many  years? 

Ans.  360  yrs.  300  da.  20  ho.  50  m.  37  sec. 

47.  How  many  barley-corns  will  reach  round  the  globe,  it 
oemg  360  degrees  ?  -Ani-.  4755801600. 

'  48.  In  running  300  miles,  how  many  times  will  a  wheel  9  ft 
2  in.  in  circumference  turn  round?  Ans.  172800. 

t  49.  How  many  seconds  in  1828  years,  allowing  365-1-  days 
to  the  year?  Ans.  57687292800. 

50.  What  is  the  cost  of  a  tun  of  wine  at  3  cents  per  gill  ? 

II  Ans.  D241.92. 

f  REVIEW. 

When  is  Compound  Reduction  used  ?  When  several  de- 
nominations are  given  to  be  reduced,  how  will  you  proceed  ? 
What  is  the  rule  ?    How  will  you  prove  questions  in  this  rule  1 


PROPORTION;  OR,  SINGLE  RULE  OF  THREE. 

The  first  idea  that  naturally  presents  itself  to  the  mind  in  the 
consideration  of  Proportion,  is  Ratio,  or  the  connexion  existing 
between  Ratio  and  Proportion ;  for  ratio  is  a  Latin  word,  and 
means  proportion,  although  it  can  only  exist  between  quantities 
of  the  same  kind.  The  numbers  given  are  called  terms,  and 
may  be  considered  as  a  divisor  and  dividend  ;  and  the  ratio,  the 
required  'term,  as  a  quotient  in  a  division  sum.  The  terms  are 
written  thus,  5  :  15,  ratio  3 ;  3  x5  =  15  ;  the  ratio  of  84  :  336 
=  4,  because  84  is  contained  in  336,  4  times  ;  so  may  any  two 
numbers  of  the  same  kind  form  a  ratio.  What  is  the  ratio  of 
9  to  18  ?  Ans.  2.  What  is  the  ratio  of  6  to  24  ?  What  is  the 
ratio  of  11  to  13  ?  Ans.  1^\.  What  is  the  ratio  of  100  to  30  ? 
Ans.  ^^.  2  :  4  ::  8  :  16  ;  here  the  ratio  is  2,  because  2  is  in 
4=:2,  and  8  is  in  16  =  2.  2:10::  12  :  60  ;  here  the  ratio  is  5, 
because  2  is  in  10=5  ;  12  in  60=5.  16  :  15  ::  48  :  45  ;  here 
the  ratio  is  reversed,  and  will  stand  thus,  -|f ,  first  and  second 
terms  ;  ||— i|.,  the  third  and  fourth  terms. 

Proportion  is  a  comparative  relation  of  one  thing,  or  number, 
to  another  ratio.  Proportion,  when  complete,  consists  of  four 
numbers  or  terms,  and  is  a  combination  of  two  equal  ratios  ; 
and  the  product  of  the  first  and  fourth  terms  (which  are  the 
extremes)  is  equal  to  the  product  of  the  second  and  third  terms 


102  proportion;  or,  stnglk  rule   of  three. 

(middle  term  or  means.)  Therefore,  it  is  evident  that  any  one 
of  the  four  terms  is  rSadily  ohtained,  when  we  have  the  other 
three  given. 

2  :  (4  ::  8)  :   16  Thus:  if  4  lbs.  of  tea  cost  8  dollars, 

4x      2x     what  will  12  lbs.  cost? 

—        —         lbs.    lbs.      I).  Here  the  price  of 

32        32  4  :   12   ::   8       the  tea  is   2   dollars 

8  per    pound  ;     conse 

—  quently,  4  pounds  will 
4)96  cost    8    dollars,     12 

—  pounds  will   cost  24 
D24  cost,      dollars,  &c. 

D.      lbs.     D. 
Again:  If  24  :   12  ::  8  :   4  lbs.  Ans. 

8  Hence  we  find,  to  obtain  either  the 

—         first  or  fourth  term,  divide  the  product 
24)96(4lb.  of    the    second    and    third    (middle) 
terms,  by  the  given  extreme,  and  the 
quotient  will   be  the  other  extreme,  or  term  sought.     And  to 
obtain  either  of  the  middle  terms,  divide  the   product  of  the 
extremes  by  the  given  middle  terms,  and  the  quotient  will  be 
the  middle  term  required.     The  fu-st  and  second  terms  of  a 
proportion  always  express  quantities  of  the  same  kind,  and  so 
likewise  do  the  third  and  fourth  terms.     Thus  : — 
lbs.        lbs.         D.         D. 
2     :      12     ::     6     :     36 
That  is,  if  2  pounds  cost  6  dollars,  what  will  12  pounds  cost? 
Ans.  36  dollars  ;  it  is  3  dollars  per  pound,  and  3  X  12  r=: 36  dol- 
lars.    This  last  example  is  the  modern  method  of  stating  Pro- 
portion, and  now  considered  the  most  correct.     The  old  method 
was  thus:   If  21b.   :  cost  6D.  ::  what  cost  12lb.   :   36D. ;  12  X 
6=72-^2=36  as  above. 

The  following  short  rules  can  often  be  used  to  advantage  : — 

1.  Divide  the  second  term  by  the  first,  multiply  the  quotient 
by  the  third,  and  the  product  will  be  the  answer. 

lbs.        D.  lbs.         D. 

5     :     10     ::    .15     :     30 

2.  Divid-e  the  third  term  by  the  first,  multiply  the  quotient  by 
the  second,  and  the  product  will  be  the  answer. 

lbs.        D.         lbs.  D. 

5     :     10     ::     15     :     30 

3.  Divide-  the  first  term  by  the  second,  and  the  third  term  by 
that  quotient,  and  the  last  quotient  will  be  the  answer. 

D.        lbs.         D.  lbs. 

10     •     5     ::     30     :      15 


proportion;   or,  single  rule  of  three.  103 

4.  Divide  the  first  term  by  the  third,  yid  the  second  term  by 
that  quotient,  and  the  last  quotient  will  be  the  answer. 


lbs. 

D.         lbs. 

D. 

15 

:     30     ::     5     : 

GENERAL    RULES. 

10 

Rule  for  stating. — 1.  Place  that  term  in  the  third  place, 
which  is  of  the  same  name  or  kind,  with  that  in  which  the 
niiswer  is  required ;  and  if  the  answer  requires  to  be  greater 
than  the  third  term,  set  the  greater  of  the  two  remaining  num- 
bers for  the  second  term,  and  the  other  number  for  the  first 
term. 

2.  But  if  the  answer  requires  to  be  less  than  the  third  term, 
set  the  less  of  the  two  remaining  numbers  in  the  second  place, 
and  the  other  in  the  first  place. 

Rule  for  working. — 1.  Reduce  the  third  term  to  the  lowest 
denomination  mentioned  in  it. 

2.  Reduce  the  first  and  second  terms  to  the  lowest  denom- 
ination mentioned  in  either  of  them. 

3.  Multiply  the  second  and  third  terms  together,  and  divide 
by  the  first,  and  the  quotient  will  be  the  answer,  in  the  samo 
name  to  which  the  third  was  reduced ;  then  bring  this  denom- 
ination into  the  answer  required. 

When  1,  or  unity,  is  one  of  the  terms,  a  formal  statement  is 
not  required  ;  it  can  be  done  by  multiplication  or  division 


1.  Multiply  the  first  and  fourth  terms  together  (extremes). 
Multiply  the  second  and  third  terms  together  (means). 

If  the  four  numbers  are  proportional,  these  products  will  be 
equal. 

2.  Divide  the  larger  of  the  first  and  second  terms  by  the  less  ; 
divide  the  larger  of  the  third  and  fourth  terms  by  the  less,  and 
the  two  ratios  will  be  equal. 

3.  Invert  the  question. 

The  terms  may  be  distinguished  generally  in  the  following 
manner  :  the  first  term  is  preceded  with  an  if,  or  supposition, 
and  the  second  by  a  demand,  "  what  will,"  "  what  cost,"  <fcc. 


104 


PROPORTION  ;    OR,    SINGLE    RULE    OF    THREE. 


QUESTIONS. 


1.  If  4  lbs.  sugar  cost  64  cts, 

what  will  8  lbs.  cost? 

lbs.  lbs.    cts. 

4  :  8  ::  64 

8 


4.  If  5  lbs.  of  cheese  cost  45 
cents,  what  will  18  lbs.  cost? 
lbs.   lbs.      cts. 
5  :   18  ::  45 
18 


4)512 

360 
45 

D1.28  Ans. 
2.  If  2  dollars  will  purchase 

5)810 

8  pounds  of  butter,  how  many- 

pounds  will  50  cents  buy  ? 

Dl.62  Ans. 

c.       c.       lb. 

5.  If  8  lbs. 

of  pork  cost  60 

2.00  :  50  ::  8 

cents,  what  will  24  lbs.  cost? 

8 

lb.    lb.       cts. 

or,  lbs.  cts.     lbs. 

8  :  24  ::  60 

8  :  60  ::  24 

200)400 

60 

24 

Ans,     2  lbs. 

8)1440 

8)1440 

3.  If  1  bushel  of  rye  cost  75 

cts.,  what  will  25  bushels  come 

D1.80  Ans. 

m. SO  Ans. 

to  at  that  price  ? 

6.  If  1  bushel  of  wheat  cost 

75  cents. 

Dl.12.5,  what  will  70  bu.  cost  ? 

25  X   bushel. 

bush.  bush.         D. 

1   :  70 

::   1.12.5 

375 

D. 1.12.5 

150 

X 

70  bush. 

D18.75  Ans. 

D78.75.0  Ans. 

7.  If  4  yards  of  cloth  cost  D9.50,  what  will  18  yards  cost? 

Ans.  D42.75. 

8.  If  1  bushel  of  clover-seed  cost  D3.25,  what  will  7  bushels 
cost  ?  Ans.  D22.75. 

9.  If  6  bushels  of  turnips  cost  D1.25,  what  will  30  bushels 
cost?  ♦  Ans.  D6.25. 

10.  When  eggs  are  18  cents  a  dozen,  what  will  47  dozen 
cost?  Ans.  D8.46. 

11.  When  eggs  are  12^  cents  per  dozen,  how  many  can  you 
have  for  11.50?  Ans.  92  dozen. 

12.  Purchased  a  barrel  of  mackerel  for  Dll.50,  on  counting 
shem  there  were  210  fish ;  what  was  the  cost  of  1  ? 

Ans.  5  cents,  4  mills.  -^ 


PROPORTION,    OH    SINGLE    RULE    OF    THREE.  105 

13.  Purchased  a  box  containing  70  hats,  for  which  I  paid 
D280,  what  was  the  cost  of  1  liat  ?  Ans.  D4.00. 

14.  If  1  pound  of  sugar  cost  11  cents,  what  will  1200  pounds 
cost?  Ans.  D132.00. 

15.  If  5  bushels  of  salt  cost  D3.12.5,  what  will  50  bushels 
cost?  Ans.  D31.25. 

16.  If  4  men  can  perform  a  piece  of  work  in  20  days,  how 
many  days  will  it  require  for  3  men  to  do  the  same  ? 

Ans.  26.666=1 . 

17.  If  7  bushels  of  corn  cost  13D.,  what  will  40  bushels  cost? 

Ans,  D74.28.5. 

18.  When  hay  is  68  cents  per  cwt.  what  is  it  per  ton  ? 

Ans.  D13.60. 

19.  When  sugar  is  11.5  cents  per  pound  how  much  is  it  per 
cwt.?  Ans.  D12.88. 

20.  WHiat  costs  a  hogshead  of  molasses,  when  it  is  selling  at 
52  cents  per  gallon  ?  Ans.  D32.76. 

21.  If  a  barrel  of  wine  cost  50  dollars,  how  much  is  that  per 
quart?  Ans.  39.7  cents  nearly. 

22.  If  a  bushel  of  cloverseed  cost  6  dollars,  what  is  that  per 
pint  ?  Ans.  9.37  cents. 

23.  If  30  men  can  perform  a  piece  of  work  in  11  days,  how 
many  men  will  it  require  to  perform  a  piece  of  work  4  times  as 
large  in  12  days  ?  Ans.  110  men. 

24.  How  many  men  must  be  employed  to  finish  a  piece  of 
work  in  15  days  which  5  men  can  do  in  24  days  ?  Ans.  8  men. 

25.  If  6  muli  can  harvest  a  piece  of  grain  in  12  days,  in 
what  time  will  24  men  do  it  ?  Ans.  3  days. 

26.  How  long  must  a  board  be,  that  is  9  inches  wide,  to  make 
12  square  feet?  9  in.  :  12  ::  12  ft.  Ans.  16  feet  long. 

27.  if  a  pound  of  sugar  cost  9.5  cents,  what  will  be  the  price 
of  a  hogshead  weighing  5  cwt.  2  qrs.  17  lbs.  ?  Ans.  D60.13.5. 

28.  If  a  person's  daily  expenses  be  D2.25,  and  his  annual 
income  1250D.,  what  sum  will  he  save  at  the  end  of  the  year? 

Ans.  428.75. 

29.  How  much  tea  can  you  purchase  with  500D.,  when  it  is 
selling  at  Dl.25  per  pound?  Ans.  400  pounds. 

30.  If  a  man  is  paying  D2.12.5  per  week  for  board,  how* 
much  is  it  for  1  year?  (52  weeks=l  year.)         Ans.  DllO.50. 

31.  If  1  cwt.  of  sugar  cost  D11.50,  what  will  15  cwt.  2  qrs. 
18  lbs.  cost  ?  Ans.  D180.09.8. 

32.  If  the  annual  income  of  a  person  is  800D.,  how  much 
may  he  use  daily,  and  have  300 D.  remaining  at  the  expiration 
of  the  year?  Ans,  D1.37  nearly. 


106  PROPORTION,    OR    SINGLE    RULE    OF    THREE. 

33.  If  4.25  yards  of  cloth  cost  D16.25,  what  will  16.5  yaiis 
cost?  Ans.  D63.08.8. 

34.  If  1.25  yards  of  sheeting  cost  12.5  cents,  what  will  50.5 
yards  cost  ?  Ans.  D5.05. 

35.  If  1.5  bushels  of  wheat  cost  D2.25,  what  will  20.5  bush- 
els  cost  ?  Ans.  D30.75 

36.  If  20.5  bushels  of  wheat  cost  D30.75,  how  much  is  it 
per  bushel  ?  Ans.  Dl.50. 

37.  How  much  will  20.5  bushels  of  wheat  cost,  at  Dl.50  per 
bushel  ?  Ans.  D30.75. 

38.  A  farmer  sold  6  bags  of  wheat  at  1.12  dollars  per  bushel 
of  60  pounds  ;  the  bags  on  being  weighed,  contained  the  follow- 
ing number  of  pounds  of  grain,  namely:  No.  1,  186  pounds; 
No.  2,  182  pounds  ;  No.  3,  176  pounds;  No.  4,  181  pounds  ; 
No.  5,  190  pounds;  No.  6,  185  pounds  ;  it  is  required  to  know 
how  many  bushels  he  had,  and  how  much  money  he  received. 

lbs.  lbs.         lbs.  D.  lbs. 

186         or  60  :   1100  ::   1.12  then  60)1100(18  bushels. 

182  1.I2X  60 

176  D.  

181  60)123200(20.53.3J  Ans.         500 

190  480 

185  

bush.  20 

60)1100(18.333=^  4xpecks. 

D.  60)80(1  peck. 

1)1.12  60 

+  18i- bushels.  IS-J  bushels.  — 

60  X  pounds.  20 

896                          8  X  quarts. 

112  1080                              

20=1  of  60  60)160(2  quarts. 

20.16                          120 

.371  price  i  bu.  1100  lbs.  proof. 


40 


D20.531  2  X  pints. 

2d  Ans.  18  bush.  1  pk.  2  qt.  1  pt.  1 J  g.  — 

60)80(1  pint. 
39.  What  is  the  cost  of  1890  pounds  60 

of  wheat,  at  1.15  dollars  per  bushel  of  60  — 

pounds?     1890  lbs. -^60=31.5  bushels  20 

XD1.15  =  D36.22.5.  Ans,  4  X  gills. 

Or,  60  :   1890  ::   1.15  Ans.  D36.22.5.  — 

60)80(lf§  =  |=^ 


PROPORTION,    OR    SINGLE    RULE    OF    THREE.  107 

40.  When  corn  is  selling  at  62.5  cents  a  bushel,  how  much 
can  you  have  for  125  dollars  ?  Ans.  200  bush. 

41.  Paid  175  dollars  for  120  bushels  of  rye,  what  did  one 
bushel  cost?  Ans.  D1.45.8  + 

42.  Bought  a  barrel  of  flour  for  11.50  dollars,  what  is  the 
cost  of  1  pound?  Ans,  5. Sets. + 

43.  What  will  16.5  bushels  of  oats  cost,  at  37.5  cents  per 
bushel?  Ans.  D6.18.75. 

44.  What  will  2  loads  of  corn  come  to,  at  75c.  per  bushel, 
one  load  has  35.5  bushels,  the  other  37  bushels  ?  Ans.  D54.37.5. 

45.  If  4  cords  of  wood  cost  15.50  dollars,  what  will  12.5 
cords  cost?  Ans.  D48.43.7. 

46.  If  7.5  acres  will  produce  87.25  bushels  of  grain,  what 
quantity  will  18.75  acres  produce  ?     Ans,  bush.  218.125=4  qts. 

47.  When  hay  is  selling  at  18  dollars  per  ton,  how  much  is 
It  per  cwt.  ?  Ans.  90  cents. 

48.  When  wood  is  selling  at  7D.  per  cord,  how  much  is  that 
per  foot  ?  Ans.  5.4  cents -f- 

49.  What  is  the  cost  of  a  stove  weighing  2  cwt.  1  qr.  18  lbs., 
at  2.50  dollars  per  cwt.  ?  Ans.  D6.02.6. 

50.  What  is  the  cost  of  1800  chestnut  rails,  at  3.75  dollars 
per  thousand  ?  Ans.  D6.75. 

51.  If  1  cwt.  of  sugar  cost  13  dollars,  50  cents,  what  will  17 
cwt.  3  qrs.  14  lb.  cost  ?  Ans.  D241.31.2. 

52.  If  1  tun  of  wine  cost  300  dollars,  what  cost  1  gallon  ? 

Ans.  1  dollar  19  cents. 

53.  If  a  hogshead  of  molasses  cost  50  dollars,  how  much 
must  it  be  sold  for  per  gallon,  to  gain  ten  dollars  on  the  cost  ? 

Ans.  95.2  cents. 

54.  Purchased  a  pipe  of  wine  for  150  dollars:  15  gallons 
leaked  out ;  how  much  must  the  remainder  be  sold  for  per  gal- 
lon, to  get  the  first  cost?  Ans.  D  1.35.1 

55.  Bought  a  cask  of  wine  at  87.5  cents  per  gallon,  and  paid 
130D.,  how  many  galls,  did  it  contain  ?  Ans.  148  galls.  2  qts.  2  gi. 

^Q,  A.  is  indebted  to  B.  965  dollars,  but  A.  compounds  with 
him,  and  agrees  to  pay  him  75  cents  on  the  dollar ;  how  much 
will  B.  receive  ?  Ans.  723  dollars,  75  cents. 

57.  If  a  man  spend  9  cents  a  day,  what  will  it  amount  to  in 
a  year  ?  Ans.  32  dollars,  85  cents. 

58.  How  much  wheat,  at  1  dollar,  15  cents,  per  bushel,  can 
you  purchase  for  125  dollars  ?         Ans.  108  bush.  2  pks.  6  qts. 

59.  In  what  time  will  48  men  perform  a  piece  of  work  which 
12  men  can  do  in  24  days  ?  Ans.  6  days, 

60»  Sound  moves  at  the  rate  of  1142  feet  per  second :  if  the 


108  PROPORTION,    OR    SINGLE    RULE    OF    THRKE. 

report  of  a  piece  of  ordnance  be  heard  1  minute,  3  seconds  after 
the  flash  was  observed,  the  distance  is  required  ?  Arts.  13m.  5fur.  • 
.■      61.  Of  what  length  must  a  board  be,  that  is  4.5  inches  in 
"wi3th,,to  make  a  square  foot  ?  Ans.  32  inches. 

62.  Ml  dollar  and  25  cents  per  week,  how  many  weeks  can 
you  board  for  150  dollars  ?  Ans.  120  weeks. 

63.  If  25  horses  will  consume  85  bushels  of  grain  in  4  weeks, 
how  many  bushels  will  8  horses  require  in  the  same  time  ? 

Ans.  27  bushels  and  6  quarts. 

64.  The  president  of  the  United  States  receives  a  salary  of 
25000  dollars  annually,  what  sum  does  he  receive  daily  ? 

Ans.  D68.49.3-I- 

65.  When  butter  is  12.5  cents  per  pound,  how  much  can  you 
have  for  137  dollars  ?  Ans.  1096  pounds'. 

66.  If  1  bushel  of  corn  cost  78   cents,  what  is  the  cost  of 
64.56  bushels  ?  .472^.  D50.35.6. 

67.  If  5  pounds  of  venison  cost  45  cents,  what  must  you  give 
for  165  pounds  ?  Ans.  14  dollars,  85  cents. 

68.  When  leather  is  selling  at  22.5  cents  per  pound,  what 
will  750  pounds  cost?  Ans,  168  dollars,  75  cents. 

69.  If  .5  yards  of  cloth  cost  37.5  cents,  what  will  1.25  yards 
cost?  Ans.  93.75  cents. 

70.  If  .75  pounds  of  indigo  cost  87.5  cents,  what  will  1.5 
pounds  cost  ?  Ans.  1  dollar  and  75  cents. 

71.  If  1  acre  of  land  cost  D18.25,  what  will  5  acres,  2  roods, 
2  poles  cost  ?  Ans.  DlOO.60.312. 

72.  If  1.5  yards  of  silk  cost  D2.50,  what  will  1  quarter,  2 
nails  cost?  Ans.  62.5  cents. 

73.  What  cost  6  cwt.  of  sugar,  at  10  cents  per  pound? 

Ans.  D67.20. 

74.  If  100  dollars  in  12  months  bring  6  dollars  interest,  what 
sum  will  bring  the  same  in  8  months  ?  Ans.  150  dollars. 

75.  How  many  yards  of  carpeting  that  is  .75  yards  wide,  are 
sufficient  to  cover  a  floor  that  is  18  feet  wide  and  60  feet  long  ? 

Ans.  160  yards. 

76.  If  8  acres  of  land  cost  D182.50,  what  will  12.7  acres 
ost?  Ans.  D289.71.9. 

77.  If  9.5  acres  of  land  cost  240  dollars,  what  is  the  value 
of  7.25  acres?  Ans.  183.15.7. 

78.  If  15  yards  of  cloth  cost  D39.45,  how  many  yards  can 
you  buy  for  21  dollars.  Ans.  7  yards,  3  qUoLrters,  3  nail* 

79.  If  95  bushels  of  corn  cost  D68.25,  what  will  320  cost  : 

Ans.  D229.89.4, 

80.  If  40  dollars  will  pay  for  14^  yards  of  cloth,  how  many 
ean  you  buy  for  75  dollars  ?   Ans.  26  yards,  2  quarters,  3  nail^ 


FKOroilTK)X  ;    OR,    SINGLE    RULE    OF    THREE.  100 

81.  If  a  car  will  run  552  miles  on  a  railroad  in  24  hours, 
w  f:ir  will  it  run  in  13  hours  ?  Arts.  299  miles. 

82.  II"  15  pounds  of  coflee  cost  Dl.80,  what  will  18  pounds 
cost?  Ans.  D2.16. 

83.  How  much  land  at  D2.50  per  acre  should  be  given  in 
exchange  for  360  acres  at  D3.75  per  acre  1        Ans.  540  acres. 

84.  How  many  feet  of  window-glass  in  a  box  containing  144 
panes,7by9?    144x7x9=9072;  12  X  12=144)9072(63  ^n^. 

85.  How  many  cubical  inches  in  a  cubical  mile  ? 

Thus,  1760  yds.  x3x  12  =  63360  inches  in  1  mile  (L.  M.) 
63360x63360x63360=254358061056000  cubical  inches  =  l 
cubical  mile.  Ans. 

86.  How  many  cubical  miles  in  the  globe  ? 

Thus,  as  355  :  113  ::  360x69.5  :  7964  diameter  of  the  earth. 
Then,  360  x  69.5  x  7964  X  1327.33=264482820122  cubical 
miles  in  the  globe.  Ans.     7964-^-6=1327.33+  —^  diam. 

87.  How  many  cubical  inches  in  the  globe  ? 

Ans.  67273337308854741368832000  inches. 

88.  If  you  purchase  7  yards  of  cloth  for  42  dollars,  how 
many  yards  can  you  have  for  600  dollars  ?  and  if  you  sell  the 
same  at  the  rate  of  40  dollars  for  6^  yards,  will  you  gain  or  lose, 
and  how  much?  Ans.  gain  D15.38.4-f 

89.  A  merchant  bought  49  tuns  of  wine  for  910  dollars,  the 
freight  cost  90  dollars,  duties  40  dollars,  cellar  31  dollars  67 
cents,  other  charges  50  dollars,  and  he  would  gain  185  dollars 
by  the  bargain ;  what  must  I  give  him  for  23  tuns  ? 

Ans.  D613.33. 

90.  The  earth,  being  360  degrees  in  circumference,  turns 
round  on  its  axis  in  24  hours ;  how  far  does  it  turn  in  1  minute 
in  the  43d  parallel  of  latitude,  the  degree  of  longitude  in  this 
latitude  being  about  51  statute  miles?  Ans.  12|  miles. 

91.  A.  and  B.  depart  from  the  same  place,  and  travel  the 
same  road,  but  A.  goes  5  days  beforelB.  at  the  rate  of  20  miles 
per  day;  B.  follows  at  the  rate  of  25  miles  per  day;  in  what 
time  and  distance  will  he  overtake  A .  ? 

Ans.  20  days ;  distance  500  miles. 

92.  If  f  of  a  yard  of  cloth  cost  D8.25,  what  will  |  yard  cost  ? 

(1=2.25)     Ans.  24.75. 

93.  If  300  barrels  of  flour  cost  5070  dollars,  what  will  200 
cost?     What  is  f  of  5070  ? 

94.  If  I  of  a  barrel  cost  -^^  of  a  dollar,  what  will  f  cost? 
What  is  HXtt  •  ^^^-  4if=42.9c.-f- 

95.  What  will  7  acres  2  roods  38  poles  of  land  cost  at 
D64.50  per  acre  ?  Ans.  D499.06.875. 

10 


110  proportion;  or,  single  rule  of  three. 

96.  What  is  the  cost  of  328  yards  of  calico  at  13J  cents  per 
yard  ?  Ans.  D43.733333^^. 

97.  What  is  the  sum  of  60  dollars  and  nine  hundredths,  12 
dollars  and  three  tenths,  18  dollars  and  three  thousandths,  and 
44  dollars  and  three  hundredths  ?  Ans.  Dl 34.423. 

98.  What  is  the  cost  of  19  gallons,  2  quarts,  1  pint,  1  gill  of 
wine,  at  3  cents  per  gill  ? 

99.  When  a  merchant  compounds  with  his  creditors  for  40 
cents  on  the  dollar,  how  much  is  A.'s  part,  to  whom  he  owes 
2500  dollars?  how  much  is  B.'s  part,  to  whom  he  owes  1600 
dollars  ?  Ans.  A.  1000  dollars  ;  B.  640  dollars. 

100.  If  30  bushels  of  rye  may  be  bought  for  120  bushels  of 
potatoes,  how  many  bushels  of  rye  may  be  bought  for  600  bush- 
els of  potatoes?  Ans.  150,  ratio  5, 

review. 

What  do  you  understand  by  ratio  ?  What  does  the  ratio  of 
two  numbers  express  ?  What  is  it  equal  to  ?  What  is  the  ratio 
of  1  to  5  ?  of  2  to  8  ?  of  6  to  36  ?  of  10  to  100  ?  of  12  to  2  ?  &c. 
What  are  we  able  to  ascertain  by  Proportion,  or  Single  Rule  of 
Three  ?  Why  is  it  called  Proportion  ?  Which  of  the  terms 
will  you  place  in  the  third  place  ?  After  having  set  down  the 
third  term,  what  will  you  do  next?  If  the  answer  ought  to 
be  greater  than  the  third  term,  how  will  you  do  ?  If  the  answer 
ought  to  be  less  than  the  third  term,  how  will  you  proceed  ? 
After  stating  the  question,  what  will  you  do  with  the  third  term  ? 
What  will  you  do  with  the  first  and  second  terms  ?  Which  of 
the  two  terms  will  you  multiply  together  for  a  dividend  ?  Which 
of  the  terms  will  be  the  divisor  ?  After  division,  in  what  de- 
nomination will  your  answer  be  ?  What  must  be  done  to  com- 
plete the  operation  ?  What  can  you  say  of  the  remainder,  if 
any  ?  How  can  you  prove  Proportion  ?  How  do  you  know 
when  questions  belong  to  Proportion?  When  1,  or  unity,  is 
one  of  the  terms,  how  can  the  question  be  solved  ?  What  is  the 
rule  for  stating  ?  What  is  the  rule  for  working  ?  Proof.  How 
many  rules  have  you  been  through  ?  Name  them.  How  many 
questions  have  you  solved  in  those  rules?  Ans,  1125.  What 
is  your  opinion  of  Proportion  ? 

Note. — All  the  rules  in  Proportion  may  be  deduced  from  the 
algebraical  expression,  a  :  b  ::  c  :  d,  meaning  that  a  divided  by 
b,  gives  the  same  quotient  as  c  divided  by  d.  A  and  D  are 
called  the  extremes,  B  and  C  the  means;  the  product  of  the 
means,  divided  by  one  extreme,  will  give  the  other  extreme  ; 
and  the  product  of  the  extremes,  divided  by  one  of  the  means, 
will  give  the  other  mean  ;  and  this  furnishes  the  various  rules 
of  Proportion. 


proportion;  or,  double  rule  of  three.  Ill 

COMPOUND  PROPORTION;  OR,  DOUBLE  RULE  OF 
THREE. 

Compound  Proportion,  or,  as  it  is  usually  called,  Double 
Rule  of  Three,  is  different  from  Simple  Proportion,  in  always 
having  an  odd  number  of  terms  given,  as  five,  seven  ;  it  teaches 
to  solve  such  questions  as  require  two  or  more  statements  by 
Simple  Proportion,,  and  hance  its  name.  The  remarks  and 
explanations  in  Simple  Proportion  are  equally  applicable  in  the 
present  case.  All  questions  in  Compound  Proportion  may  be 
solved  by  the  following  rules  : — 


1.  Place  that  term  in  the  third  place,  which  is  of  the  same 
name  with  that  in  which  the  answer  is  required. 

2.  Then  with  this  third  term,  and  each  pair  of  similar  terms, 
finish  the  statement  as  in  the  preceding  directions  in  the  Single 
Rule  of  Three,  and  set  them  in  the  first  or  second  places,  as 
directed  for  stating  in  that  rule. 

3.  Reduce  the  third  term  to  the  lowest  denomination  men- 
tioned in  it,  and  the  other  term  to  similar  denominations. 

4.  Multiply  the  first  two  terms  together  for  a  divisor,  then 
multiply  the  two  terms  in  the  second  place,  and  that  product  by 
the  third  term  for  a  dividend. 

5.  Divide,  and  the  quotient  is  the  answer  in  the  same  de- 
nomination with  the  third.     Proof,  invert  the  terms. 

Note. — There  are  several  methods  by  which  the  rules  of  pro- 
portion may  be  cancelled  or  contracted ;  for  sufficient  reasons, 
those  rules  will  be  omitted  here. 

EXAMPLES. 

1 .  If  4  men  in  12  days  can  harvest  48  acres  of  grain,  how 
many  acres  can  8  men  harvest  in  16  days  ? 
Thus:  if  men,  4  :  8  >    .    .g  Explanation. — In  the 

days,  12    16  J  "  *     first  place,  we   are  told 

that  4  men  in  1 2  days  can 

divisor — 48  128  harvest  48  acres  of  grain, 

48  12x4=48  days;  this  is 

just  1  acre  per  day  for 

48)6144(128  acres   Ans,  each  man,  because  they 
or,  16x8  =  128  acres.  labor  48  days,  and  har- 

vest 48  acres  ;  consequently,  if  8  men  labor  16  days,  they  will 
harvest  128  acres,  because  16  x  8  =  128  acres.  The  product  of 
4  and  12  bears  the  same  proportion  to  48  acres  as  the  product 


113  proportion;    or,    double    RULli    OF    THREE. 

of  8  and  16  does  to  the  answer,  128,  which  in  this  case  are 
the  same ;  then,  if  we  multiply  the  product  of  8  and  16  by  48 
acres,  the  5th  term,  and  divide  by  the  product  of  4  and  12,  tho 
result  or  proportion  will  still  be  the  same,  &;c. 

2.  If  lOOD.  in  12  months  will  gain  7D.  interest,  how  much 
will  750D.  gain  in  48  months  ? 


lOOD.     :     750D.  ::   >  7D.     or  750D. 
12  mo.         48  mo.     >    •  7x  per  cent. 


1200  36000  5250 

7D.  4x  years. 


12,00)2520,00(210  Ans.     D210.00  Ans. 
If  lOOD.  in  12  months  will  gain  7D.  then  it  follows  that  in 
the  same  proportion  750D.  will  gain  210D.  in  48  months. 

3.  I  lOOD.  in  365  days  will  gain  6D.,  how  much  will  GOOD 
gain  in  95  days  ? 

DlOO     :     900D.  ::    )  6 
365  X  da.  95da. 


36500        8550 
6X 

D.  c. 

365)513000(14.05.4+  Ans. 

4.  If  400D.  will  gain  in  7  months  14D.,  what  is  the  rate  per 
cent,  per  annum  ?  Ans.  6D. 

5.  If  lOOD.  will  gain  6D.  in  a  year,  in  what  time  will  400D. 
gaiti  14D.?  Ans.  7  months. 

6.  If  16  men  can  moAv  112  acres  of  grass  in  7  days,  how 
many  acres  can  24  men  mow  in  19  days  ?  Ans.  456. 

7.  If  2  men  can  construct  12  rods  of  wall  in  6  days,  how 
many  rods  can  8  men  build  in  24  days  ?  Ans.  192. 

8.  If  5  men  can  make  300  pairs  of  shoes  in  40  days,  how 
many  men  will  it  take  to  make  900  pairs  in  60  days  ?    Ans.  10. 

9.  If  4D.  will  hire  8  men  3  days,  how  many  days  must  20 
men  work  for  40D.  ?  Ans.  12. 

10.  If  4  men  receive  12D.  for  3  days'  work,  how  many  men 
will  it  require  to  earn  48D.  in  16  days  ?  Ans.  3. 

11.  If  100  bushels  of  oats  be  sufficient  for  18  horses  20  days, 
how  many  bushels  will  60  horses  require  in  36  days  ? 

Ans.  600  bushels, 

12.  If  the  transportation  of  8  cwt.  cost  D12.80  for  128  miles, 
what  sum  must  be  paid  for  the  carriage  of  4  cwt.  32  miles  ? 

Ans.  D1.60 


PROPORTION,    OR    DOUBLE    RULE    OF    THREE.  113 

13.  How  many  dollars  will  it  require  to  gain  6D.  in  1  year 
if  wuth  560D.  1  gain  56D.  in  1  year  and  8  months  ?  Ans.  lOOD. 

14.  If  6  tons  of  hay  be  sufficient  for  8  horses  7  months,  how 
much  will  serve  20  horses  1  year  and  5  months  ? 

Ans.  36  T.  8  cwt.  2  qrs.  8  lbs. 

15.  If  10  horses  in  18  days  consume  the  grass  of  2  acres,  how 
many  acres  will  20  horses  require  in  27  days  ?      Ans.  6  acres. 

16.  If  20  bushels  of  wheat  are  sufficient  for  a  family  of  8 
persons  5  months,  how  much  will  be  sufficient  for  4  persons  12 
months  ?  Ans.  24  bushels. 

17.  If  8  men  can  build  a  wall  20  feet  long,  6  feet  high,  4  feet 
thick,  in  12  days,  in  what  time  will  24  men  build  one  200  feet 
long,  8  feet  high,  and  6  feet  thick  1 

84-12=20x6  divisor.     200x6x8  dividend. 

18.  If  a  family  of  8  persons  use  360D.  in  nine  months,  how- 
much  will  serve  a  family  of  18  persons,  12  months  ? 

Ans.  1080U. 

19.  If  lOOD.  in  52  weeks  gain  6D.  interest,  how  much  will 
200D.  gain  in  26  weeks  1  Ans.  6D. 

20.  If  350D.  in  six  months  gain  DlO.50  interest,  what  will  be 
the  interest  of  400D.  for  4  years?  Ans.  96D. 

21.  If  a  family  of  6  persons  expend  300D.  in  8  months,  how 
much  will  serve  a  family  of  15  persons  for  the  same  time  ? 
Thus:  per. per.      D.        D. 

6  :   15  ::  300  :  750  Ans.  solved  by  a  single  statement. 

22.  If  750D.  will  support  a  family  of  15  persons  for  8  months, 
how  much  will  serve  them  20  months  ? 

mo.  mo.        D.         D.     or      8  mo.  :  20  ::  >  750  :  1875. 
8  :  20  ::  750  :   1875 -4;i^.  15  per.    15      5  Ans. 

The  number  of  persons  (15)  maybe  omitted  in  statements  of 
this  kind. 

REVIEW. 

What  is  Compound  Proportion  ?  When  you  are  about  to 
make  a  statement  in  this  rule,  which  of  the  terms  is  first  to  be 
,  set  down  ?  In  what  place  ?  What  is  then  to  be  considered  ? 
What  is  to  be  done  with  the  two  terms  which  stand  in  the  first 
place  ?  What  is  to  be  done  with  the  two  terms  which  stand  in 
the  second  place  ?  By  what  do  you  multiply  the  product  of  the 
two  terms  standing  in  the  second  place,  and  by  what  do  you 
divide  that  product  for  the  answer,  when  they  consist  of  differ- 
ent denominations  ?     Repeat  the  rule. 

10* 


114  VULGAR    FRACTIONS. 

VULGAR  FRACTIONS. 

Remarks. — A  fraction  is  either  vulgar  or  decimal,  and  as  the 
word  implies,  is  a  broken  number,  or  parts  ;  a  proper  fraction  is 
less  than  unity.  Those  parts  may  be  expressed  by  figures,  as 
well  as  whole  things  ;  a  whole  is  called  an  integer,  but  a  part, 
or  some  parts  of  a  thing  are  denoted  by  figures  as  one  third,  one 
fourth,  seven  tenths,  &;c.,  of  a  thing;  the  expression  of  those  parts 
of  figures  are  called  fractions.  The  term  fraction  is  derived  from 
a  Latin  word  which  signifies  to  break,  as  an  integer  or  unity  is 
supposed  to  be  broken  or  divided  into  a  certain  number  of  equal 
parts,  one  or  more  of  which  parts  are  denoted  by  the  fraction ; 
thus,  one  fourth  (1)  denotes  one  of  the  four  equal  parts,  &c.,  into 
which  a  thing  is  broken,  or  integer  divided.  We  may  also  view 
it  as  a  part  of  a  certain  number  of  units  ;  -J  may  either  be  con- 
sidered as  two  thirds  of  one  or  one  third  of  two,  for  one  third 
of  two  is  the  same  quantity  as  two  thirds  of  one,  consequently 
if  the  numerator  of  a  fraction  be  viewed  as  an  integer,  and  divi- 
ded into  as  many  equal  parts  as  the  denominator  indicates,  the 
fraction  may  be  regarded  as  expressing  one  of  those  parts  ;  thus, 
if  4  be  divided  into  5  equal  parts,  the  fraction  }  would  express 
one  of  them,  and  the  fraction  |-  would  express  four  of  them, 
40-^5  =  8x4=32  +  8=40;  8x5  =  40,  &c.  Fractions  natu- 
rally arise  from  the  operation  of  division,  when  the  divisor  is 
not  contained  a  certain  number  of  times,  exactly,  in  the  dividend. 
For  the  remainder,  after  the  division  is  performed,  is  a  part  of  the 
dividend  which  has  not  been  divided,  the  divisor  being  the  num- 
ber of  parts  into  which  the  integer  is  divided,  and  the  remainder 
showing  the  number  of  those  parts  expressed  by  the  fraction. 
Thus  4  is  contained  in  9  twice  and  one  fourth  times,  and  hence  the 
quotient  can  not  be  fully  expressed  in  such  cases,  except  by  a 
whole  number  and  a  fraction.  As  before  observed,  fractions  are 
of  two  kinds,  vulgar  and  decimal  (the  latter  have  already  been 
explained)  which  may  be  reduced,  or  changed  from  one  to  the 
other,  still  retaining  the  same  value  ;  thus,  f ,  which  means  three 
parts  or  three  quarters  of  anything,  would,  if  expressed  deci- 
mally, be  .75,  or  -^^q,  which  is  the  same,  namely,  three  quarters. 

Vulgar  fractions  are  expressions  for  any  assignable  parts  of 
a  unit  or  whole  number,  and  are  represented  by  two  numbers 
placed  one  above  another,  with  a  line  drawn  between  them,  thus  : 
|,  |,  &c.  The  number  above  the  line  is  called  the  numerator, 
and  that  below  the  line  the  denominator.  The  denominator 
shows  how  many  parts  the  integer  is  divided  into,  and  the  nu- 
merator shows  how  many  of  those  parts  are  meant  by  the  frac* 


VULGAR    FRACTIONS.  115 

tion.  If  an  apple  or  anything  be  divided  into  4  equal  parts,  one 
part  is  called  ^ ;  if  divided  into  two  equal  parts,  each  part  is 
called  ^f  two  of  which  =1  ;  if  divided  into  three  equal  parts, 
one  part  is  ^,  two  of  those  parts  |,  two  thirds  and  one  third  =1, 
when  divided  into  eight  equal  parts,  one  part  is  ^,  which  is  the 
half  of  ^,  two  eighths  =i,  four  eighths  =^,  six  eighths  =^, 
seven  eighths  and  one  eighth  =1  ;  -jgg-  of  a  dollar  is  3  cents  : 
Jfj^Q  is  70  cents  ;  ^^  is  6-J-  cents  ;  |  is  12^  cents  ;  -J-  is  25  cents  ; 
Y  is  l^-  dollars ;   \^  is  2  dollars,  &c. 

DEFINITIONS. 

Fractions  are  either  proper,  improper,  single,  compound,  or  mixed, 

1.  A  Single  or  Simple  Fraction  is  a  fraction  expressed  in  a 
simple  form,  as  ^,  |-,  y^g,  &c. 

2.  A  Compound  Fraction  is  a  fraction  expressed  in  a  com- 
pound form,  being  a  fraction  of  a  fraction,  as  -^  of  J,  ^  of  ^^  of 
^^,  which  are  read  thus  :  one  half  of  three  fourths,  two  sevenths 
of  five  elevenths  of  nineteen  twentieths. 

3.  A  Proper  Fraction  is  a  fraction  whose  numerator  is  less 
than  its  denominator,  as  |-,  |,  &c.  Less  than  unity,  because 
the  numerator  is  less  than  the  denominator. 

4.  An  Improper  Fraction  is  a  fraction  whose  numerator  ex- 
ceeds its  denominator,  as  -I,  |-,  ^y^,  &c.  More  than  unity,  be- 
cause the  denominator  is  less  than  the  numerator. 

5.  A  Mixed  Number  is  composed  of  a  whole  number  and  a 
fraction,  as  7|-,  35^3,  (fee,  that  is,  seven  and  three  fifths,  &c. 

6.  A  Complex  Fraction  is  one  that  has  a  fraction  for  its 
numerator,  or  for  its  denominator,  or  for  both  its  terms,  as — 
z  ^  el 

2.'  |.'  _i  ^  Sic.     An  equal  or  even  fraction  is  f  ==1,  f  =1,  &c. 

7.  A  fraction  is  said  to  be  in  its  lowest  or  least  term  when  il 
is  expressed  by  the  least  number  possible. 

8.  The  common  measure  of  two  or  more  numbers  is  that  num- 
ber which  will  divide  each  of  them  without  a  remainder ;  thus,  5 
is  the  common  measure  of  10, 20,  and  30  ;  and  the  greatest  num- 
ber which  will  d(»this  is  called  the  greatest  common  measure. 

9.  A  number  which  can  be  measured  by  two  or  more  num- 
bers is  called  their  common  multiple  ;  and  if  it  be  the  least  num- 
ber which  can  be  so  measured,  it  is  called  the  least  common 
multiple,  thus,  40,  60,  80,  100,  are  multiples  of  4  and  5;  but 
their  least  common  multiple  is  20. 

Note. — The  product  of  two  or  more  numbers  is  a  common 
multiple  of  those  numbers,  thus,  3x4x5=60;  and  60,  or 
3x4x5,  is  evidently  divisible  without  a  remainder  by  each  of 
those  numbers,  and  the  same  must  be  true  in  every  similar  case. 


116  VULGAR    FRACTIONS. 

10.  A  prime  number  is  one  which  can  be  measured  only  by 
itself  or  a  unit,  as  3,  7,  23,  &c. 

1 1 .  A  perfect  number  is  equal  to  the  sum  of  all  its  aliquot 
parts.  An  aliquot  part  of  a  number  is  contained  a  certain  num- 
ber of  times  exactly  in  the  number. 

The  following  perfect  numbers  are  all  which  are  at  presen 

known : — 

6  8589869056 

28  137438691328 

496  2305843008139952128 

812  2417851639228158837784576 

33550336         9903520314282971830448816128 
Before  proceeding  farther,  it  will  be  proper  to  introduce  the 

two  following  problems,  which  will  be  found  very  useful  and 

important  in  the  reduction  and  solution  of  questions  involving 

vulgar  fractions : — 

PROBLEM    I. 

To  find  the  greatest  common  measure  of  two  or  more  numbers. 

RULE. 

1.  If  there  be  two  numbers  only,  divide  the  greater  by  the 
less,  and  this  divisor  by  the  remainder,  and  so  on,  always  divi- 
ding the  last  divisor  by  the  remainder,  till  nothing  remains,  then 
will  the  last  divisor  be  the  greatest  common  measure  required. 

2.  When  there  are  more  than  two  numbers,  find  the  greatest 
common  measure  of  them  as  before  ;  then  of  that  common  meas- 
ure and  one  of  the  other  numbers,  and  so  on,  through  all  the 
numbers,  to  the  last ;  then  will  the  greatest  common  measure, 
last  found,  be  the  answer. 

3.  If  1  happens  to  be  the  common  measure,  the  given  num- 
bers are  primes  to  each  other,  and  found  to  be  incommeasurable, 
or  in  their  lowest  terms. 

EXAMPLES. 

1.  What  is  the  greatest  common  measure  of  1836,  3996,  and 
1044? 

1836)3996(2.    So  108  is  the  greatest  com.  meas.  of  3996, 1836 
3672  Hence  108)1044(9 

972 

324)1836(5  

1620  72)108(1 

— —  72 

216)324(1  — 

216         (last  greatest  com.  meas.  36)72(2 

72 

Common  meas.    108)216(2  — 

216    (Therefore  36  is  the  ans.  required. 


VULGAR    FRACTIONS.  117 

2.  Find  the  greatest  common  divisor  of  the  two  number^  ^« 
*nd  81.  63)81(1 

63 

18)63(3 
54 

Greatest  common  divisor,      9)18(2 

18 

3.  What  is  the  greatest  common  measure  of  1224  and  1080  ? 

Ans.  72. 

4.  What  is  the  greatest  common  measure  of  1440,  672,  and 
3472?  Ans.  16. 

5.  What  is  the  greatest  common  divisor  of  492, 744,  and  1044  ? 

Ans.  12. 

6.  Find  the  greatest  common  divisor  of  24,  48,  and  96. 

Ans.  24. 

PROBLEM    II. 

To  find  the  common  multiple  of  two  or  more  numbers, 

DEFINITION. 

Multiple  is  a  number  which  contains  another  several  times, 
as  9  is  the  multiple  of  3,  containing  it  3  times  ;  16  of  4,  &c. 

RULE     I. 

Divide  by  any  number  that  vi^ill  divide  two  or  more  of  the 
given  numbers  without  a  remainder,  and  set  down  the  quotients, 
together  with  the  undivided  numbers,  in  a  line  beneath. 

2.  Divide  the  second  line  as  before,  and  so  on,  till  there  are 
no  two  numbers  that  can  be  divided  ;  then  the  continued  product 
of  the  divisors  and  quotients  will  give  the  multiple  required. 

EXAMPLES. 

1.  What  is  the  least  common  multiple  of  6,  10,  16,  and  20  ? 

Explanation. — I    survey    my    given 

numbers,  and  find  that  5  will  divide  two* 

of  them,  namely,  10   and  20,  which  I 

divide  by  5,  bringing  into  a  line  with  the 

quotients,  the  numbers  which  5  will  not 

measure :   Again,  I  view  the  numbers 

3       1     *4       1  in  the  second  line  and  find  two  will 

*****  measure  them  all,  and  get  3,  1,  8,  2,  in 

5y  2  X2  x3x4=240     the  third  line,  and  find  that  2  will  meas- 

Ans.)  ure  8  and  2,  and  ia  the  fourth  line  get 


*^)Q 

10     16 

20 

*2)6 

2     16 

4 

♦2)3 

1        8 

2 

118  VULGAR    FRACTIONS. 

3,  1,  4,  1,  all  prime;  I  then  multiply  the  prime  numbers  and 
the  divisors  continually  into  each  other  for  the  number  sought, 
and  find  it  to  be  240,  answer. 

2.   What  is  the  common  multiple  of  6,  3,  and  4  1 
Operation.     3)6  .  3  .  4 


2)2  .   1   .  4 

1.1.2,     3X2X2  =  12  Ans, 

3.  Find  the  least  common  multiple  of  3,  12,  and  8.    Ans.  24. 

4.  Find  the  least  common  multiple  of  2,  7,  14,  and  49. 

Ans.  98. 

5.  What  is  the  least  common  multiple  of  6  and  8  ?    Ans.  24. 

6.  What  is  the  least  number  that  3,  5,  8,  and  10,  will  meas- 
ure ?  Ans.  120. 

7.  What  is  the  least  number  which  can  be  divided  by  the  9 
digits,  separately,  without  a  remainder?  Ans.  2520. 

RULE    II. 

Divide  the  number  by  any  prime  number,  which  will  divide 
it  without  a  remainder ;  then  divide  the  quotient  in  the  same 
way,  and  so  continue  until  a  quotient  is  obtained  which  is  a 
prime.  Then  will  the  successive  divisors,  together  with  the 
last  quotient,  form  the  prime  factors  required.  Select  all  the 
different  factors  which  occur,  observing,  that  when  the  same 
factor  has  different  powers,  to  take  the  highest  power.  The 
continued  product  of  the  factors  thus  selected  will  give  the  least 
common  multiple. 

8.  What  is  the  least  common  multiple  of  12,  16,  and  24  ? 
These  number,  resolved  into  their  prime  factors,  give — 

12=22x3 

16=2* 

24=2^X3 
Therefore,  2x2  =  4x2  =  8x2=16x3=48.  Ans. 
Note. — This  rule  is  considered  the  most  correct. 
'  Note, — Any  number  ending  with  an  even  number  or  cipher  is 
divisible  by  2 ;  any  number  ending  with  5  or  0  is  divisible  by  5. 
If  the  right-hand  place  of  any  number  be  0,  the  whole  is  divisible 
by  10.     If  the  two  right-hand  figures  of  any  number  be  divisible 
by  4,  the  whole  is  divisible  by  4.     If  the  three  right-hand  figures 
of  any  number  be  divisible  by  8,  the  whole  is  divisible  by  8. 

INTRODUCTORY    QUESTIONS. 

If  you  pay  50  cents  for  ^  of  a  yard,  what  will  1  yard  cost  ? 


REDUCTION    or    VULGAR    FRACTIONS.  119 

If  1  yard  cost  ID.,  what  will  1  part  cost?  If  1^  yards  cost  3 
dimes,  what  is  1  yard  worth  ?  If  2|  yards  cost  4D.,  what  is  \ 
of  a  yard  worth  ?  If  1  yard  cost  2^D.,  what  is  the  cost  of  |-  of 
a  yard  ?  If  1  bushel  of  wheat  cost  l^D.,  what  did  you  pay  for 
1  peck  ?  If  2j  bushels  cost  S^D.,  what  cost  1  bushel  ?  How 
many  are  6  thirds?  How  many  are  15  thirds?  How  many 
are  24  eighths  ?  How  much  is  f ,  i/,  ^^,  J-|-  of  a  dollar  ?  At 
y^^  of  a  dollar  per  yard,  what  will  4  yards  cost  ?  What  cost  | 
of  a  yard,  at  Dl'^  per  yard  ?     How  much  is  3  times  f  ? 


REDUCTION  OF  VULGAR  FRACTIONS. 

The  reduction  of  fractions  is  bringing  them  from  one  form 
into  another,  in  order  to  prepare  them  for  the  operations  of  ad- 
dition, subtraction,  &c. 

To  abbreviate,  or  reduce  fractions  to  their  lowest  terms. 


Divide  the  terms  of  the  given  fractions  by  the  least  number . 
which  will  divide  them  without  a  remainder,  and  the  quotients 
again  in  the  same  manner,  and  so  on,  till  it  appears  that  there  is 
no  number  greater  than  1  which  will  divide  them,  and  the  frac- 
tion will  be  in  its  lowest  terms.  Or,  divide  both  the  terms  of 
the  fraction  by  their  greatest  common  measure,  and  the  quotients 
will  be  the  terms  of  the  fraction  required. 

Reduce  |-||  to  its  lowest  terms. 
(4)     (3) 

Thus:  8)f||=:f§=y«^=|.  A;i^.    Or,  288)480(1 

288 

192)288(1 
192 

Common   measure,  96 
Then  96)f||=|.  Ans.,  as  before. 
96  is  the  greatest  common  measure,  and  f  the  answer. 

1.  Reduce -3-2_5o  to  its  lowest  terms.  ^)2|,  ^)^§§,  5)|,  ^)2|=r-J.  ' 

2.  Reduce  i|-  to  its  lowest  terms.  Ans.  j^j, 

3.  Reduce  If  to  its  lowest  terms.  }, 

4.  Reduce  j2_4_  to  its  lowest  terms.  | 

5.  Reduce  ^^  to  its  lowest  terms.  ^. 


120  REDUCTION    OF    VULGAR    FRACTIONS. 

6.  Reduce  -^^jr  to  its  lowest  terms.  Ans,  | 

7.  Reduce  jYA  ^^  ^^^  lowest  terms.  |- 

8.  Reduce  jjW  to  its  lowest  terms.  ^^. 

9.  Reduce  ||-  to  its  lowest  terms.  |. 
10.  Reduce  g-^-|-  to  its  lowest  terms.  ^, 

To  reduce  an  improper  fraction  to  a  whole  or  mixed  number. 

RULE    II. 

Divide  the  numerator  by  the  denominator,  and  the  quotient 
will  be  the  whole  number.  If  there  be  any  remainder,  set  it 
over  the  given  denominator,  for  the  numerator  of  the  fraction. 

11.  Reduce  ^-^  to  its  proper  terms.     25-^6—  Ans.  4^, 

12.  Reduce  '^^-^  to  its  proper  terms.  7^^^. 

13.  Reduce  %^  to  its  proper  terms.  8. 

14.  Reduce  ^/  to  its  equivalent,  or  whole  number.  9. 

15.  Reduce  '^j-^^  to  its  equivalent,  or  mixed  number.  127j*^. 

To  reduce  a  7nixed  number  to  its  equivalent  improper  fraction, 

RULE    III. 

Multiply  the  whole  number  by  the  denominator  of  the  frac- 
tion, and  add  the  numerator  of  the  fraction  to  the  product,  under 
which  subjoin  the  denominator,  and  it  will  form  the  fraction 
required. 

16.  Reduce  19i|-  to  an  improper  fraction. 

19xl8z:r342  +  12i=354.    ^  Ans.^^^. 

17.  Reduce  lOOJf  to  an  improper  fraction.  ^M^- 

18.  Reduce  514^^^  to  an  improper  fraction.  ^xf^* 

19.  Reduce  127Y^y  to  an  improper  fraction.  ^ff^* 

To  reduce  a  compound  fraction  to  a  simple  one. 

RULE    IV. 

Multiply  all  the  numerators  together  for  a  new  numerator,  and 
all  the  denominators  for  a  new  denominator. 

20.  Reduce  t  of  |-  of -J  to  simple  fractions. 

1X2X3  =  6  :  2X3X4=24  :  ^\=1 

21.  Reduce  |^  of  f  of  f  to  a  simple  fraction.  -g^. 

22.  Reduce  i  of  |  of  f  to  a  simple  fraction.  -jj. 

23.  Reduce  f  of  |  of  ^|  to  a  simple  fraction.  f^. 

To  reduce  fractions  of  different  denominations  to  equivalent  fraC' 
tions  having  a  common  denominator. 

RULE    V. 

Multiply  each  numerator  into  all  the  denominators  except 


REDUCTION    OF    VULGAR    FRACTIONS.  121 

its  own,  for  a  new  numerator,  and  all  the  denominators  into  each 
other,  continually,  for  a  common  denominator. 

24.  Reduce  ^,  f ,  and  |,  to  equivalent  fractions  having  common 
denominators.  Thus,  Ix5x8=:40,  new  numerator  for  ^  : 
2x4x8=64  for  |  :  5x4x5  =  100,  num.  for  f  :  4x5x8  = 
160,  common  denominator.     Thus  the  equivalent  fractions  are 

Reduce  ^,  f ,  and  f,  to  a  common  denominator. 

Thus,  7x5x7=245  :  4x9x7=252  :  3x5x9=135  : 
9X5X7=315.  Ans,§immi. 

The  least  common  denominator  may  be  found  by  dividing 
both  the  terms  of  a  fraction  by  any  numbers  that  will  make  the 
denominators  alike,  for  a  common  denominator.  Thus  jq^^q  and 
i^250=f ,  i ;  -8%  and  ^\^^=^\  and  /^. 

To  reduce  a  fraction  of  one  denomination  to  an  equivalent  frac- 
tion of  a  higher  denomination,  retaining  the  same  value. 

RULE    VI. 

Multiply  the  given  denominations  by  the  parts  in  the  several 
denominations  between  it  and  that  denomination  to  which  it  is 
to  be  reduced,  for  a  new  denominator,  which  is  to  be  placed 
under  the  given  numerator, 


25.  Reduce  I-  of  a  cent  to  the  fraction  of  a  dollar. 


i_ 
-5* 


7Xt^o=to-o=^^^-tK 

26.  Reduce  -g-  of  a  pwt.  to  the  fraction  of  a  lb.  iroy.      tfjo 

27.  Reduce  \\  of  a  mill  to  the  fraction  of  a  dollar.      y^Voo- 

28.  Reduce  f  of  a  mill  to  the  fraction  of  an  eagle.      -nr^TTo- 

29.  Reduce  |  of  a  lb.  avoir.,  to  the  fraction  of  a  cwt.       ji^. 

To  reduce  a  fraction  of  one  denomination  to  an  equivalent  fraction 
of  a  lower  denomination,  retaining  the  same  value, 

RULE    VII. 

Multiply  the  given  numerator  by  the  parts  in  the  denomina- 
tions between  it  and  that  denomination  you  would  reduce  it  to, 
for  a  new  numerator,  which  place  over  the  given  denominator. 

This  rule  is  the  very  reverse  of  rule  6  ;  they  will  prove  each 
other. 

30.  Reduce  ^ts  ^^  ^  dollar  to  the  fraction  of  a  cent. 

^1^  of  y  of  V> ;  then  ^i^  X  \o  =  \o  =  m=Ans.  j. 

31.  Reduce  ^  of  a  lb.  avoir.,  to  the  fraction  of  an  oz.  y. 

32.  Reduce  j^  of  a  cwt.  to  the  fraction  of  a  lb.  avoir.       ^. 

33.  Reduce  J5V00  ^^  ^  dollar  to  the  fraction  of  a  mill.      J^' 

34.  Reduce  ytth  ^f  ^  ^^*  ^^^7'  ^  ^^^  fraction  of  a  pwt,   |  pwt, 


122  REDUCTION    OF    VUl^OAR    FRACTIONS. 

To  find  the  value  of  a  fraction  in  tjie  known  parts  of  the  integer 
as  of  coins,  weights,  measures,  <5^c, 

RULE    VIII. 

Multiply  the  numerator  by  the  parts  of  the  next  less  denom- 
ination, and  divide  the  product  by  the  denominator  ;  and  if  any- 
thing remains,  multiply  by  the  next  less  denomination,  and  di- 
vide by  denominator  as  before,  and  so  continue,  as  far  as  is  neces- 
sary, and  the  quotients  placed  after  one  another  will  be  the  answer. 

35.  Reduce  -f  of  a  dollar  to  its  proper  quantity. 

Thus,  4  multiplied  by  100—5=^^^   80  cents. 

36.  Reduce  If  of  a  day  to  its  proper  quantity.  6  hours. 

37.  Reduce  -^-^  of  an  eagle  to  its  proper  quantity.      D  1.87. 5. 

38.  What  is  the  value  of  |  of  a  lb.  troy?  7  oz.  4  pwt. 

39.  What  is  the  value  of  f  of  a  yard  ?  2  qrs.  2|-  na. 

40.  What  is  the  value  of  f  of  a  mile  1         6  fur.  26  po.  1 1  ft. 

41.  What  is  the  value  of  j^-  of  a  dollar  ?  43c.  7im. 

42.  What  is  tho  value  off  of  an  acre  ?  3  R.  17-1-  po. 

43.  What  is  the  value  of  y^  of  a  day  ?     16  h.  36  m.  55/3  s. 

To  reduee  any  given  quantity  to  the  fraction  of  any  greater  de- 
nomination, of  the  same  kind, 

RULE    IX. 

Reduce  the  given  sum  to  the  lowest  denomination  mentioned 
for  a  numerator,  and  the  denomination  of  which  to  make  it  a 
fraction,  to  the  same  for  a  denominator. 

44.  Reduce  6dimes,2  cents,  and  5  mills,  to  the  fraction  of  a  D. 

6—2—5  X  10  X  10  =  625— Ans.  62J  cents 

45.  Reduce  D  1.87. 5  to  the  fraction  of  an  eagle.  -^^ 


46.  Reduce  c43.7|m.  to  the  fraction  of  a  D.  -j^^ 


47.  Reduce  7  oz.  4  pwt.  to  the  fraction  of  a  lb.  troy 

48.  Reduce  6  fur.  26  po.  11  ft.  to  the  fraction  of  a  mile.     -I 

49.  Reduce  3  R.  17^  po.'  to  the  fraction  of  an  acre.  |. 

To  reduce  a  whole  number  to  an  equivc^lent  fraction,  having  a 
given  denominator, 

RULE    X. 

Multiply  the  whole  number  by  the  given  denominator ;  place  the 
product  over  said  denominator,  and  it  forms  the  fraction  required 

50.  Reduce  6  to  a  fraction  whose  denominator  shall  be  8. 
Thus,  6x8=48  and  ^^  Ans,      Proof:   4.8=48-^8=6 

51.  Reduce  15  to  a  fraction  whose  denominator  shall  be  12 

A71S,  \y^, 

52.  Reduce  100  to  a  fraction  whose  denominator  shall  be  70. 


ADDITION    OP    VULGAR    FRACTIONS.  123 

To  reduce  a  given  fraction  to  another  equivalent  one^  having  a 
given  numerator, 

RULE     XI. 

The  numerator  of  the  given  fraction  is  the  first  term ;  the 
numerator  of  the  proposed  fraction,  the  second  term  ;  the  de- 
nominator of  the  given  fraction  is  the  third  term,  to  the  denom- 
inator required. 

53.  Reduce  f  to  a  fraction  of  the  same  value,  the  numerator  of 
which  shall  be  15.     Thus,  as  3  :  16  ::  4  :  20  denom.  Ans.  \^. 

54.  Reduce  |  to  a  fraction  of  the  same  value  having  its  nu- 
merator 42.  Ans.  \^, 

55.  Reduce  f  to  an  equivalent  fraction,  the  numerator  of 
which  shall  be  27.  Ans.  ^-^3. 

To  reduce  a  given  fraction  to  another  equivalent  one,  having  a 
given  denominator. 
RULE   xir. 
As  the  denominator  of  the  given  fraction  is  to  the  denom- 
inator of  the  intended  fraction,  so  is  the  numerator  of  the  given 
fraction  to  the  numerator  required. 

56.  Reduce  |;  to  an  equivalent  fraction,  having  its  denomina- 
tor 24.         As  8  :  24  ::  7  :  21  numerator.  Ans.  |i 

57.  Reduce  |-  to  a  fraction  of  the  same  value,  the  denomina- 
tor of  which  shall  be  45.  ,        Ans.  |-|-. 

58.  Reduce  ^j  to  an  equivalent  fraction,  having  its  denom- 
inator 68.  Ans.  1|-- 

59.  Reduce  |  to  a  fraction  of  the  same  value,  the  denomina- 
tor of  which  shall  be  46.  Ans.  34^. 

46 

60.  Reduce  ^j  to  an  equivalent  fraction,  having  its  denom- 
inator 20.  Ans.  12  8  ^ 

5f  .  20^^ 

61.  Reduce  the  mixed  fraction  zrr  to  a  simple  fraction,  thus, 
5|=2-^  and  7^=^|,  then  it  becomes  a  complex  fraction, 
A.,  then  23x11=253  and  85x4=340.  ^ns.^^^. 

" # 

ADDITION  OF  VULGAR  FRACTIONS. 

RULE. 

1.  If  it  is  necessary,  reduce  the  given  fractions  to  simple 
fractions ;  that  is,  compound  fractions  to  single  ones ;  mixed 
numbers  to  improper  fractions  ;  fractions  of  different  integers  to 


124        SUBTRACTION  OF  VULGAR  FRACTIONS. 

those  of  the  same ;  and  all  of  them  to  a  common  denominator , 
then  the  sum  of  the  numerators,  written  over  the  common  de- 
nominator, will  be  the  sum  of  the  fractions  required. 

2.  After  the  fractions  are  prepared,  multiply  each  numerator 
into  all  the  denominators  but  its  own,  and  take  their  sum  for  a 
new  numerator ;  multiply  all  the  denominators  for  a  new  de- 
nominator. 

61.  What  is  the  sum  of  ^^  of  4|,  f  of  -^,  and  Q-J  ? 

Ans.  i^-^^  =  12^^. 

First,  4|=  V,  and  9-|  =  ^ ,  and  /^  of  ^  =  2^5_9  ^nd  |  of  ^=|. 

After  reducing  them  to  improper  fractions,  they  will  stand  thus  : 

2_5_9    3    ^^A  si      Thpn 
8  0  '  8'  **"^    4  •       -*^-iAt;ii, 

259  X    8x4==z8288  Explanation. — After    the 

3  X  80  X  4=  960  ,  fractions  are  prepared  or  re- 

37  X  8  X  80r::23680  duced,  according  to  the  pre- 

ceding  rules,  you  multiply 

32928  each  numerator  into  all  the 

^=:i|^9  =  12|-§    denominators  but  its    own, 

80  X  8  X  --^==2560  and  take  their  sum  for  a  new 

numerator;  then  multiply  all 

the  denominators  for  a  new  denominator.  Sic. 

62.  Add  together  ^^,  |,  and  A.  Thus,  2x9x4=72; 
8xl3x4=i416;   1x9x13  =  117;  13x9x4=468;  ^j^n, 

72  +  117+416=605  ) 

\  —113  7      Anv 

468     468     468     468  ) 

63.  Add  f  and  y^^  together. 
5)5  10  10->   5x2=4)  4  +  5  =  9 


:)  4+5=9 

>num. Ans. 

)  10  10  10 


1     2  com.  denom.  10-f- 10x5=5. 

64.  Add  ^-Q,  ^^,  and  |,  together.        .  Ans.  2^^-^. 

65.  Add  f  and  17^  together.  18^. 

66.  Add  iD.  |c.  ^^^c.  and  |  mills  together.  20c.  9.m. 

67.  Add  f ,  9i  and  f  of  i  together.  9^gi. 

68.  Add  f  of  a  dollar  to  -|  of  a  dollar.  Dl.12.5. 

69.  Add  4  of  a  ton  to  ^^  of  a  cwt. 

Ans.  12  cwt.  1  qr.  8  lbs.  12  oz.  12f  dr. 


SUBTRACTION  OF  VULGAR  FRACTIONS. 

RULE. 

1.  Prepare  the  fractions,  when  necessary,  as  in  Addition,  and 
ake  the  difference  of  the  numerators,  under  which  write  the 


MULTIPLIOATION    OF    VULOAR    FRACTIONS.  125 

comraoa  denominator,  which  will  give  the  difference  of  th« 
fraction  required. 

2.  Multiply  each  numerator  by  the  other  denominator,  and 
subtract  the  less  (product)  from  the  greater,  for  the  numerator 
sought ;  multiply  one  denominator  by  the  other  for  a  common 
denominator. 

70.  From  }|  take  f     15x7=105;   4x16=64;  16x7= 

n2;if|-A%=TV2  ^^^- 

71.  From  5  take  y\.     1  X  14=14  common  denominator. 

14—     1X5  =  70   >   7^         8   _62_->l  6        A„r, 

14^14x8=  8  (T^T^-Tl-^T^  •^^^• 


72.  From  96i  take  14|.'      c-rn      /}  v^y-v.,^    Ans.  81y^ 
75.  From  8^  take  5|  i^  '       •' 


74.  From  f  take  f .         yf.    r)    /* 

7fi     From  8-1  take  .51.     ''       ^  ' 


Tod! 


76.  From  5i  take  f .  4 

77.  From  12  take  f.  11 

78.  From  |  take  |.  | 


MULTIPLICATION   OF  VULGAR  FRACTIONS. 


RULE. 

Reduce  compound  fractions  to  simple  ones,  and  mixed  num- 
bers to  improper  fractions ;  if  an  integer  be  given,  reduce  it  to 
an  improper  fraction,  by  putting  a  unit  for  its  denominator ;  then 
multiply  the  numerators  together  for  a  new  numerator,  apd  thtp 
denominators  together  for  a  new  denominator. 

Note, — In  some  cases,  when  the  nmnerators  and  denomina- 
tors are  equal,  they  may  be  omitted 


79.  Multiply 
f  Ans. 

6|  b|  8. 

Thus,  6|=V; 

t^n 

Vx|=='p-- 

80. 

Multiply 

|bf|. 

81. 

Multiply 

0^7  by 

1- 

Ans.  1| 

82. 

Multiply 

H  by  9^. 

Ans.  69|- 

83. 

Multiply 

\  by  5|. 

5*=¥ 

;  |xy 

=H= 

=44^  Ans. 

84. 

Multiply 

Sibyf 

Ans.  I. 

126  MULTIPLICATION    OF    VULGAR    FRACTIONS. 

To  multiply  a  whole  number  by  a  fraction,  or  a  fraction  by  a 
whole  number. 


Multiply  the  whole  number  by  the  numerator  of  the  fraction, 
and  divide  the  product  by  the  denominator ;  if  the  numerator  be 
1,  divide  by  the  denominator  only ;  if  there  is  a  whole  number 
multiply  by  it,  and  add  the  products. 

85.  86.  87.  88. 

Multiply  15     or  15  35  68  42 

by     3-1-  3i  5f 


7^        45  11§ 

45  7^        175 


m 

9f 

12)748 

4)126 

62^ 
.    476 

31* 

378 

52^A.    52^         186fA 

538^A.  409^A. 
89.  At  30  cts.  per  bushel,  what  will  l^j-^  bushels  cost^ 
Thus:         19tV  15)30 

30  — 


5.70 

2 


D5.72  Ans. 

90.  How  many  are  50  roods,  multiplied  by  5|  ? 
50x5^=250;  50-^-2=25  +  250=275  Ans. 

91.  What  will  22 j\  of  cloth  cost  at  11  dollars  per  yard? 
245.  What  will  22^?  246.  What  will  23^^^  260.  W^hat 
willS,-^?  42.  What  will  31^?  351.  What  will  99y\  ?  1095. 
An;?.  2239  dollars.         Thus,     22^^         11)11 

11  — 

1X3=3 

245  +  3 

92.  If  a  man's  salary  be  1200  dollars  a  year,  what  will  2^-^ 
years  come  to?<|2420.  What  will  3/^?  3630.  What  will 
5^2.o_?  6040.  What  will  8|§f?  10701.  What  will  12||^? 
15598.     Ans.  38398  dollars. 

93.  If  lib.  of  butter  cost  ^^  of  a  dollar,  what  will  205lb8.  cost? 

3  X  205=615  ) 

Thus,         —  \  =30i|D.  Ans. 

20  20  S 


DIVISION    OF    VULGAR    FRACTIONS.  127 

94.  What  cost  400  yards  of  calico,  at  |  of  a  dollar  per  yard  ? 
3x400=1200)  ^ 

>  =D150.  Ans,  /s2,    . 

How  much  IS  13  times  fi^r  ^  ^*^  Ans.  3|||. 

How  much  is  513  times  y\?  ^{\^  =326/y. 

Multiply  3^  by  367.  1192f 

Multiply  6|  by  211.  1450|. 

99.  Multiply  3/^  by  42.  1291. 

100.  How  much  is  314  multiplied  by  f  ?  235^. 

101.  How  much  is  513  multiplied  by  y^^.?  3263-^. 

102.  Multiply  42  by  \^.  11. 


;-> 


DIVISION  OF  VULGAR  FRACTIONS. 


Prepare  the  fractions  as  before  ;  then  invert  the  divisor,  and 
proceed  exactly  as  in  multiplication.  The  product  will  be  the 
quotient  required. 

103.  Divide  5^  by  7f .     Thus,  ^^^,  and  7f =3_i  5  then  ^ 
multiplied  by  ^  =if|=li-  ^'^^' 

104.  Divide  84  by  i|f.  84  multiplied  by  320=26880-t- 
168=160.  Ans. 

In  the  last  example  I  multiplied  the  dividend  by  the  denomi- 
nator of  the  dividing  fraction,  and  then  divided  the  product  by 
the  numerator. 

105.  Divide  100  by  2f.  Ans,  36 /y. 

106.  Divide  \\  by  |.  3)i7^8^^l2|.  ^^^. 

107.  Divide  4  by|.  Ans.  4^ 
'  108.  Divide  4|  by  f  of  4.  2  " 

109.  Divide  99  by  108. 

110.  Divide  5i  by  7f. 

111.  Divide!  by  9. 

112.  Divide  92  by  4f  *  20|. 

rule  II. 

Divide  the  numerator  by  the  whole  number,  writing  the  de- 
nominator under  the  quotients  ;  or,  multiply  the  denominator  by 
th©  whole  number,  writing  the  result  under  the  numerator. 


^1- 

I 


128  APPLICATION    OF    VULGAR    FRACTIONS. 

113.  If  8  yards  cost  jg-D.,  how  much  per  yard  ? 

i^g-:-8z=:yL.D.=-A7i5.  6\  cenis 

114.  Divide  I  by  8.  =z-^\=z 

115.  Divide  5-^  by  6.  =/o  = 

116.  Divide  6-|  dollars  among  5  men. 

^4 —   4      •    ^ — 20" 

117.  Divide  16f  by5. 

118.  Divide  Sf  by  6. 

119.  Divide  114^  by  280. 

120.  Divide  620  by  8^-2- . 

8y2_=:ff  :  620xll=6820-^90=75Jg= 

121.  Divide  92  by  41. 

122.  Divide  86  by  15|-. 

123.  How  many  square  rods  in  1210  square  yards  ? 
30i  sq.  rods=i|i   :   1210  X4==4840-^121 

124.  Divide  f  by  j\.         f  x  V^|f = 

125.  Divide  If  by^6_, 

126.  How  many  times  are  -^^  contained  in  f  ^  ? 

127.  Divide  9J  by  J  of  |.  '37. 


% 

Ans.  l^V 

Ans.  75J. 
20f 

? 

=Ans.  40. 

2. 

.» 

APPLICATION  OF  VULGAR  FRACTIONS. 

128.  What  part  of  1  month  is  19  days  ?  Ans,  if. 

129.  What  part  of  1  month  is  25  days,  13  hours  ?  fif . 

130.  What  is  the  value  of  f  of  a  cwt.  ?  1  =  3  qrs. 

131.  What  is  the  value  of  fif  of  a  hhd.  ?     49  galls.  1 29/3  qt. 

132.  What  is  the  value  of  -^^  of  a  day  ?  16  h.  36  m.  55^^^  sec. 

133.  Reduce  y||y  ^^  ^  minute  to  the  fraction  of  a  day.  1:^. 

134.  Reduce  f  of  a  pint  to  the  fraction  of  a  hhd.     2"sVo=-6^o* 

135.  What  part  of  a  yard  is  3  qrs.  3  na.  ■?  i|. 

136.  Reduce  |  and  |  to  a  common  denominator.         ^^,  |-|. 

137.  Reduce  ^  of  f ,  and  |,  to  a  common  denominator.  ^^,  |^. 

138.  How  much  are  5^  times  5^  ?  30^. 

139.  How  much  are  16^  times  16^  ?  272i. 

140.  Multiply  ^\  of  ^\  by  f  tV4\- 

141.  Divide  8f  by  6.  fl  =  lH' 

142  How  much  is  60  times  j|q  ?  f =2i' 

143  Redace  J  of  |  of  |  to  a  single  fraction.  ^. 
144.  Reduce  f  of  |  of  5  to  a  single  fraction                       21 . 


SIMPLE    PROPORTION    IN    VULGAR    FRACTIONS. 


12d 


145.  Reduce  ^  off  of  |  to  a  single  fraction.  Ans,  ^j, 

146.  Add  74,  f  off,  and  7,  together. 

First,  7f  =Y,  Y  of  |=i|-  and  7=^  ;  then  the  fractions  are 
3_9^l|^  and  |;  therefore,  39 x 56x1  ==2184  ;  15x5x1  =  75; 

7  X  5X56  Xr::^  1960:  + =:4219-r-5x  56x1  ==289;  =  152VV^^^- 
Or  thus:         2184x75x1960 


280  =:152fo^^^. 

147.  From  5  take  ^.  Ans.  4^^» 

148.  Multiply  7]  by  8-|  ;  6|  by  6f . 

Y  multiplied  by  2_7_7^2_9^45_9_  .  ^^  45.5625.  Ans. 

149.  From  |J  take  f.     40x6=240;  5x41=205;  41x6 
z=:246  ;  ^^Q      246^^24"6'  Ans, 

1150.  From  fl  take  j\,  Ans.  j^^, 

151.  From  I- take  f.  i-|. 

152.  From  ll|  take  f .  1/^. 

153.  From  }  of  j^^  take  ^  of  ^^.  .^Vo- 

154.  Add  together  -j?,  |,  and  i.  li||. 

155.  Add  together  14f  and  15f .  30^. 


SIMPLE  PROPORTION  IN  VULGAR  FRACTIONS. 

RULE. 

1.  Reduce  the  given  numbers,  if  necessary  ;  mixed  numbers 
to  improper  fractions ;  like  terms  to  the  same  denominations, 
and  each  to  their  lowest  terms. 

2.  Then  state  the  question  as  in  whole  numbers  ;  invert  the 
first  term,  and  multiply  all  the  numerators  together  for  a  new 
numerator,  and  all  the  denominators  for  a  new  denominator,  and 
divide  the  upper  term  by  the  lower  for  the  answer.  Or  you  can 
reduce  the  given  fractions  to  decimals,  and  work  by  the  general 
rule  of  proportion. 

1.  If  |-  of  a  yard  of  cloth  cost  D2f ,  what  will  5^^  yards  cost  ? 

Thn«  •      'S  5    65.    r)95._2_l   .    t}.p„     8  y  65  y  21  =  10920 6_5 

inUS.      Oj.^_j2,    -L'^8^—    8     >    ^"^"    T'^T^'^   ¥=    ^72     —   4» 

65-i-4  =  Dl6.25;  |  :  ff  ::   V  ;  Ans.  D161. 

2.  How  much  silk  that  is  f  of  a  yard  wide,  will  line  6|  yards 
of  cloth,  which  is  1^  yards  wide? 


4» 


130  SIMPLE    PROPORTION    IN    VULGAR    FRACTIONS. 

1.  Common  way: — 

1|-  yds.  :  6f  yds.  ::  3  qrs.  ;  then  5  qrs.  :  27  qrs.  ::  3  qrs. 
X4  X4  X5 

2.  By  vulgar  fractions  : —    f) 

First,  l\=l  6f  =  V»  and  3  qrs.rz:|  ;  then,  as  f  :  2/   ..   3 
^nd^X'-i  Xi=ix\'  X^=%\^  =  '-^  =  Ui  yds.  Ans. 

3.  By  decimals  : — 

11=1.25,  6f  =  6.75,  andf =.75;  then  as  1.25  :  6.75  ::  .75, 
(inverse.)  Ans.  11.25  yards. 

3.  How  much  in  length,  that  is  13|-  poles  in  breadth,  will 
make  a  square  acre?.  Ans.  llj^^i-. 

4.  If  a  suit  of  clothes  can  be  made  of  4|  yards  of  cloth,  1| 
yards  wide,  how  many  yards  of  cloth  |-  of  a  yard  wide  will  it 
require  for  the  same  person  ?  Ans.»6  yds.  1  qr.  3f  na. 

5.  If  I  of  a  yard  cost  |-D.,  what  will  401  yards  cost  ? 

Ans.  D59.06J. 

6.  A  merchant  sold  5^  pieces  of  cloth,  each  containing  12§ 
yards,  at  12f  cents  per  yard,  what  did  the  whole  amount  to  ? 

Ans.  D8.82f . 

7.  If  f  of  a  yard  of  cloth  cost  |-  of  a  D.,  what  will  2J  yards 
cost?  Ans.  D4|l. 

8.  If  14i  yards  cost  19^D.,  how  much  will  19|-  yards  cost? 

9.  If  D9  will  buy  76-1-  lbs.  of  sugar,  how  much  will  D17|  buy  ? 

10.  If  D155  will  buy  3^  acres,  how  many  acres  will  D1576 
buy? 

11.  If  ^  of  f  of  a  gallon  cost  f  D.,  what  will  5^  gallons  cost  ? 

Ans.  D91. 

12.  If  f  oz.  of  gold  be  worth  D1.50,  what  is  the  cost  of  1  oz.  ? 

Ans.  Dl.80. 


COMPOUND  PROPORTION  IN  VULGAR  FRACTIONS. 

RULE. 

Prepare  the  terms,  if  necessary,  then  state  them  agreeably  to 
the  directions  given  for  whole  numbers ;  invert  the  dividing  terms 
and  multiply  the  upper  figures  continually  for  the  numerator, 
and  those  below  for  the  denominator  of  the  fractional  answer. 


COMPOUND  PROPORTION  IN  VULGAR  FRACTIONS.     131 

IS.  If  3|-  yards  of  cloth,  |  of  a  yard  in  width,  cost  D7|,  what 
fvill  9f  yards  cost,  |  of  a  vard  in  width  ? 

Thus,  D7f = V>  9j-=:f  yd,  3^=1  yd. ;  then  if  7  .  3_9  7  ^ 
1=  ^1  divisor.  \^  X I  X  V  =  Hir  --  ¥  =  ¥3t¥  =  V¥  =D24. 
78. 1|  f  yd.  :  I  yd.  ::  D^,  or  invert  the  terms,  |xf  X\^  X 
iX  V  =  V9V-V¥=  .  ^^^-  D24.78.1-1. 

14.  If  a  footman  travel  294  miles  in  Of  days  of  12|^  hours 
long,  how  many  days  of  10-|  hours  will  it  require  him  to  travel 
762  niiies  ?  Ans.  2\^  days 

15.  If  D50  in  5  months  gain  02^-^4,  what  time  will  D13J- 
require  to  gain  Dly^^?  Ans.  9  months. 

16.  A.  and  B.  are  on  opposite  sides  of  a  circular  field,  268 
poles  about ;  they  begin  to  go  round  it,  both  the  same  way  and 
at  the  same  time ;  A.  goes  22  rods  in  2  minutes,  and  B.  34  rods 
in  3  minutes  ;  how  many  times  do  they  go  round  the  field  before 
the  swifter  overtakes  the  slower?         Ans.  A.  travels  16^ times 

[round=4422  po.  B.  17  times=4556  po. 

17.  If  A.  can  do  a  piece  of  work  alone  in  7  days,  and  B.  in 
12  days  ;  let  them  both  go  about  it  together,  in  what  time  will 
they  finish  it?  Ans.  4^-^  days. 

18.  When  12  persons  use  1|-  pounds  of  tea  per  month,  how 
much  should  a  family  of  8  persons  provide  for  a  year  ? 

Ans.  9  lbs. 

19.  If  5  persons  drink  7|-  gallons  of  beer  in  a  week,  what 
quantity  will  serve  8  persons  22^  weeks  ?         Ans.  280|  galls. 

REVIEW    OF    VULGAR    FRACTIONS. 

What  is  a  fraction  ?  What  is  a  vulgar  fraction  ?  Can  frac- 
tions be  expressed  in  various  forms  and  retain  the  same  value  ? 
What  is  common  measure  ?  How  will  you  find  the  greatest 
common  measure  of  two  numbers  only  ?  How  will  you  find 
the  greatest  common  measure  when  there  are  more  than  two 
numbers  ?  How  will  you  find  the  common  multiple  of  two  or 
more  numbers?  Explain  the  operation.  What  is  the  least 
common  multiple  of  two  or  more  numbers  ?  What  is  the  rule 
for  the  least  common  multiple?  When  is  a  fraction  in  its  lowest 
terms  ?  How  will  you  reduce  a  fraction  to  its  lowest  terms  ? 
What  are  the  lowest  terms  of  nine  twelfths  ?  What  is  an 
improper  fraction  ?  How  will  you  reduce  an  improper  fraction 
to  a  whole  or  mixed  number  ?  What  is  a  mixed  number  ? 
How-will  you  reduce  a  mixed  number  to  its  equivalent  improper 
fraction  ?  What  is  a  compound  fraction  ?  How  will  you  re- 
duce a  compound  fraction  to  a  simple  one  ?  How  will  you  re  - 
duce  ^  fraction  of  one  denomination  to  an  equivalent  fraction  of 


132  MISCELLANEOUS    MATTER  IN  VULGAR    FRACTIONS. 

a  higher  denomination  ?  How  will  you  reduce  a  fraction  of 
one  denomination  to  an  equivalent  fraction  of  a  lower  denomina 
tion  ?  How  will  you  find  the  value  of  a  fraction  in  the  known 
parts  of  the  integer  ?  How  will  you  reduce  a  given  quantity  to 
the  fraction  of  any  greater  denomination  of  the  same  kind? 
How  will  you  reduce  a  whole  number  to  an  equivalent  fraction, 
having  a  given  denominator  ?  How  will  you  reduce  a  given 
fraction  to  another  equivalent  one,  having  a  given  numerator  ? 
How  will  you  reduce  a  given  fraction  to  an  equivalent  one,  hav- 
ing a  given  denominator  ?  What  is  the  rule  for  performing  ad- 
dition of  vulgar  fractions  ?  What  will  you  do  when  a  mixed 
number  is  given  ?  What  is  to  be  observed  when  different  de- 
nominations are  given  ?  when  all  the  denominations  are  alike  ? 
Add  I  of  a  yard  to  f  of  an  inch.  Ans.  f|-|  yards,  or  14j\- 
inches.  Add  |-  of  a  cwt.  8-|  lbs.  and  3j^q  oz.  together.  Ans.  2 
qrs.  17  lbs.  12f^  oz.  How  do  you  perform  subtraction  of  vul- 
gar fractions  ?  when  the  numerator  is  the  greater  ?  What  is  to 
be  done  when  the  fractions  are  of  different  denominations  ? 
From  I  of  a  league  take  j^q  of  a  mile.  Ans.  1  m.  2  fur.  16  po. 
Repeat  the  rule  for  performing  multiplication  of  vulgar  fractions. 
How  will  you  prepare  the  given  terms  1  Required  the  product 
of  4^,  f  of  i,  and  18|.  Ans.  9y|^.  What  is  the  rule  for  per- 
forming division  of  vulgar  fractions  ?  Divide  14|  oi  ^  by  3 J 
of  6.,  Ans.  ^^Q.  How  will  you  prepare  the  fractions  for 
stating  in  simple  proportion  ?  How  will  you  then  proceed  ? 
What  is  the  rule?  How  will  you  prepare  the  fractions  for 
stating  in  compound  proportion  ?  Repeat  the  rule.  The  ques- 
tions may  be  continued  by  the  teacher. 


MISCELLANEOUS  MATTER  IN 
VULGAR  FRACTIONS. 

1.  If  7  lbs.  of  coffee  cost  |i  of  a  dollar,  what  is  that  per  lb.? 

1^  -f-7=^5~  of  a  dollar.  Ans. 

2.  If  6  bushels  of  wheat  cost  4|  dollars,  what  is  it  per  bush- 
1?  41=11=1-1=81^  cents  Ans. 

3.  How  many  times  is  3  contained  in  462^  ?         Ans.  154i. 

4.  If  a  man  spend  f  of  a  dollar  per  day,  how  much  will  he 
spend  in  8  days  ?  I  ^^  8  =  ^  =  5  dollars.  Ans. 

A  fraction  can  be  multiplied  by  multiplying  the  nwnerator, 
without  changing  the  denominator. 


I 


MISCELLANEOUS  MATTER  IN  VULGAR  FRACTIONS.     133 

5.  At  g^Tj  of  a  dollar  for  1  pound,  what  will  12  pounds  cost? 

g5_  X  12  — 60 _ 6. —.3  Q^  a  dollar.  Ans, 

6.  What  will  9J§  tons  of  hay  come  to  at  17  dollars  per  ton? 

Ans.  164^L  dollars. 

At  f  of  a  dollar  per  yard,  what  cost  7f  yards  ? 


Ans.  6^i  dollars. 
Reduce  \,  |,  and  |-,  to  fractions  having  the  least  common 
denominator,  and  add  them  together.     Thus,  H+A"+l4~4i 
r=li|-  amount. 

9.  What  is  the  amount  of  ^  of  |  of  a  yard,  ^,  and  \  of  2  yards  ? 
Reduce  the  compound  fraction  to  a  simple  fraction,  thus  : — 

^  of  1=1,  and  i  of  2=1 ;  then,  |+|+|- =i^=l^=^  yds. 

10.  -Reduce  -^^  of  a  hogshead  to  the  fraction  of  a  gallon. 

11.  Reduce  -^-^  of  a  bushel  to  the  fraction  of  a  quart. 
2.  Reduce  ^^g^  of  a  day  to  the  fraction  of  a  minute. 

(33.  Reduce  -^-^-^  of  a  cwt.  to  the  fraction  of  a  pound. 
14.  Reduce  y^Vo  ^^  ^  hogshead  to  the  fraction  of  a  pint. 
*  5.  Reduce  -^-^-^  of  a  furlong  to  the  fraction  of  a  yard. 

6.  Reduce  -^^  of  a  gallon  to  the  fraction  of  a  hogshead. 

7.  Reduce  ^^  of  a  quart  to  the  fraction  of  a  bushel. 

18.  Reduce  ^l^-  of  a  minute  to  the  fraction  of  a  day. 

19.  Reduce  ^^  of  a  pound  to  the  fraction  of  a  cwt. 

20.  Reduce  y/g^  of  a  pint  to  the  fraction  of  a  hhd. 

21.  At  6|  dollars  per  barrel  for  flour,  what  will  y^g  of  a  bar- 
rel cost  ?  Ans.  2\^\  dollars 

22.  At  D2i  per  yard,  what  cost  6|-  yards?         Ans.  D14|-|. 

23.  At  D4f  per  yard,  how  many  yards  of  cloth  may  be 
bought  for  D37  ?  Ans.  8^^  yards. 

24.  How  many  times  is  ^^  contained  in  6  ?  Ans.  f  of  1 . 

25.  If  4|  pounds  of  butter  serve  a  family  1  week,  how  many 
weeks  will  36|  pounds  serve  them  ?  Ans.  8y|  ^  weeks. 

26.  How  many  times  is  f  contained  in  4|?  Ans.  llf 

27.  Multiply  f  of  2  by  ^  of  4=3.  Ans.       Multiply  1000000 
by|=555555|.  Ans. 

28.  A  man  bought  27  gallons,  3  quarts,  1  pint,  of  molasses  ; 
what  part  is  that  of  a  hogshead  ? 

29.  From  4  days  7|  hours,  take  1  day  9y^g-  hours. 

Ans.  2  days,  22  hours,  20  minutes. 

30.  If  6-^  yards  of  cloth  cost  3  dollars,  what  cost  91  yards  ? 

Ans.  D4.269, 

31.  If  4  oz.  of  silver  cost  i^  of  a  dollar,  what  cost  1  oz.  ? 

Ans.  D1.283, 
,^^2.  If  T^  of  a  ship  cost  251  dollars,  what  is  3^  of  her  worth  ? 
H  Ans.  D53.785. 

m 


134  MISCELLANEOUS  MATTER  IN  VULGAR  FRACTIONS. 

33.  Add  together  |E.  |D.  and  licts.       Ans.  7D.  53c.  2fm 

34.  Take  3lcts.  from  l  of  2^  dollars.  Ans.  43lcts. 

Note. — The  reason  of  the  rules,  both  for  Addition  and  Sub- ' 
traction  of  Fractions,  is  manifest.  If  the  given  fractions  have 
the  same  denominator  and  are  of  the  same  denomination,  the 
sum  of  the  numerators  written  over  the  given  denominator,  will 
be  the  sum  of  the  fractions.  A  fraction  is  subtracted  from  a 
whole  number,  by  taking  the  numerator  of  the  fraction  from  its 
denominator,  and  placing  the  remainder  over  the  denominator, 
then  taking  one  from  the  whole  number,  thus : — 

From  123  12.2  12 

Take    7|  7|  f 

H  4|  llf 

Note. — To  multiply  by  a  fraction,  whether  the  multiplicand 
be  a  whole  number  or  a  fraction,  as  before  observed,  you  can 
divide  by  the  denominator  of  the  multiplying  fraction,  and  mul- 
tiply the  quotient  by  the  numerator,  thus:  20  multiplied  by  ^, 
the  product  is  15.  To  divide  by  a  fraction,  whether  the  div- 
idend be  a  whole  number  or  a  fraction,  we  multiply  by  the  de- 
nominator of  the  dividing  fraction,  and  divide  the  product  by  the 
numerator,  thus  :  20  divided  by  |-  is  26|-,  and  12  divided  by  | 
is  16,  &c.  You  will  observe  in  multiplication,  the  multiplier 
being  less  than  unity,  or  1,  will  require  the  product  to  be  less 
than  the  multiplicand ;  and  in  division,  the  divisor  being  less 
than  unity  or  1 ,  it  will  be  contained  a  greater  number  of  times, 
and  consequently  will  require  the  quotient  to  be  greater  than  the 
dividend,  to  which  it  will  be  equal  when  the  divisor  is  1,  and 
less  when  the  divisor  is  more  than  1 . 

By  Vulgar  and  Decimal  Fractions, 

35.  If  2-|  bushels  sow  an  acre,  how  many  acres  will  22 
bushels  sow  ? 

Thus,  2|  X  4+3  =  y  improper  fraction ;  22  x  4~  1 1 =8  acres 
Ans.     Or,  22-^2.75  =  8  Ans. 

36.  At  4 1  dollars  per  yard,  how  many  yards  of  cloth  may  be 
bought  for  37  dolls.  ?  Thus,  4f  x5  +  2  =  ^  ;  37x5-^22  =  8^ 
yards  Ans.     D37~4.40  =  8  yards,  1  quarter,  1  ndl\-\-Ans. 

37.  If  14|  yards  cost  75  dollars,  how  much  per  yard  ? 
Thus,   14f  x8  +  3  =  i-i^;    75  x8  =  600-^115=5jy.^rz:^V  Ans 
Or,  D75~- 14.375  yards  =  D5.2i.7  Ans. 

38.  How  many  times  is  |  contained  in  746  1 

Thus,  746 X 8-^-3==  1989i;  746~375  =  1989^lf  =--i   Ans. 


MISCELLANEOUS    MATTER    IN    YULOAR    FRACTIONS.  135 

39.  Divide  ^  by  ^=1 ;  divide  f  by  i  =  3  ;  divide  I  by  -J  =2. 

40.  If  f  of  a  yard  cost  5  dollars,  how  much  per  yard  ? 

41.  If  f  of  a  bushel  cost  3  dollars,  how  much  for  3  bushels  ? 
how  much  for  4-  a  bushel  ? 

A9       Aflfl    2     4     i_7  .     r.AA     2         3_      4         5    _14   .   ^AA    1,2     9      18 

4^.  Add  -g,  -g,  g  —  3  ,  aaa  -^-^^  y^,  j^,  jg— jg  ,  aaa  -g  ,  ^,  j^j 
¥t^  together. 

43.  There  are  3  pieces  of  cloth,  one  containing  7-|  yards, 
another  13^  yards,  and  the  other  15|-  yards,  how  many  yards 
in  all  1  First  reduce  the  fractional  parts  to  their  least  common 
denominator,  thus  ; — 

Qv9vft 4R  48    I    32156 1  3_6 — 0_8_  —  1 

^X-iXO_40  6'4    1^64    '    6'4—   64   — -^dJ— 8* 

1x4x8=32         Then,  7|-f       -^  "^ -^ 

7x4x2z=56  13f|         /TO 


4x2x8^64 


13H 
15M 


37^=1  Ans.  371 

Adding  together  all  the  64ths,  namely,  48,  32,  and  56,  you 
have  136,  that  is  "^-^-f —2-^-^—\ -,  write  down  the  fractions  g^^ 
under  the  other  fractions,  and  carry  the  two  integers  to  the 
other  integer,  and  you  have  37|-.  Ans. 

Or,  thus:  7.75  +  13.5  +  15.875  =  37.125=:|.  The  last  method 
is  the  most  convenient. 

44.  From  |  of  an  ounce  take  |-  of  a  pwt.  Ans.  11  pwt.  3  grs. 

45.  From  4  days,  7^  hours,  take  1  day  9j^  hours. 

Ans.  2  days,  22  hours,  20  min. 

46.  A  man  purchased  -^^  of  7  cwt.  of  sugar,  how  ijiuch  did 
he  purchase  ?    ,  >^/t.     2,7^^  ^  )  i ,     1  .  J  f-  /•    7  -^ , 

ATI.  14  hours,  48  minutes,  54|^  seconds,  is  what  part  of  a 
day  ? 

48.  Add  Jib.  troy  to  ^-^  of  anoz.     Ans.  6  oz.  11  pwt.  16  grs 

49.  At  3|-  dollars  per  cwt.,  what  will  6|  lbs.  cost  ? 

50.  If  |-  of  a  yard  cost  -§•  of  a  dollar,  what  will  -^^  of  a  yard 
cost? 

51.  Add  together  f  D.,  j-\  cents,  \  mills. 

52.  Add  f  of  a  yard,  f  of  a  foot,  and  ^|  of  a  mile,  together. 

53.  Add  1  of  a  week,  f  of  a  day,  \  of  an  hour,  and  \  of  a 
minute,  together. 

54.  From  5  weeks,  2|  days,  9f  hours,  take  1\  days,  141^ 
hours. 

55.  Multiply  f  of  f ,  by  f  of  i  of  1  If.  Ans.  /^. 

56.  What  is  the  continual  product  of  7,  J,  f  of  f ,  and  3^  ? 

Ans.  2|J 


136  PRACTICE. 

57.  In  a  certain  school  -^^  of  the  pupils  study  Greek,  ^^ 
Latin,  |-  arithmetic,  ^  read  and  write,  and  20  attend  to  other 
studies  ;  required  the  number  of  pupils.  Ans.  100. 

53.  In  342f  gallons,  how  many  i  of  a  gallon  ? 

Ans.  ^Y^  of  a  gallon,  or  1371  quarts. 

Note. — Any  whole  number  may  be  made  an  improper  frac- 
tion by  drawing  a  line  under,  and  putting  a  unit  or  1  for  a  de- 
nominator, as  5  may  be  expressed  thus,  |^,  and  10,  ^j,  &c. 

Reduce  .275  to  a  vulgar  fraction. 

Thus:  j2_7_5__i.25  com.  m.=ii.  Ans. 

SECOND  REVIEW  OF  VULGAR  FRACTION. 

What  do  you  mean  by  common  divisor?  by  the  greatest 
common  divisor  ?  How  can  you  multiply  a  fraction  by  a  whole 
number  ?  How  will  you  multiply  a  mixed  number  ?  When 
the  multiplier  is  less  than  a  unit,  what  is  the  product  compared 
with  the  multiplicand  ?  How  do  you  multiply  a  whole  number 
by  a  fraction  ?  How  can  you  multiply  one  fraction  by  another, 
and  can  you  do  it  in  more  than  one  way  ?  How  do  you  multi- 
ply a  mixed  number  by  a  mixed  number  ?  How  can  you  di- 
vide a  fraction  by  a  whole  number  ?  What  is  understood  by 
common  denominator  1  the  least  common  denominator  ?  Do 
cases  frequently  occur  in  vulgar  fractions,  where  the  terms  may 
be  reduced,  and  solved  by  decimals  ?  What  can  you  say  of  the 
importance  of  a  correct  knowledge  of  all  kinds  of  fractions  ? 


PRACTICE. 

RACTiCE Is  a  contraction  of  simple  proportion,  when  the  first 
"    lerm  happens  to  be  a  unit,  or  one,  and  has  its  name  from  its  daily 
}j    use  among  merchants  and  tradesmen  ;  being  an  easy  and  con- 
7    cise  method  of  working  many  questions  which  occur  in  trade  and 
business ;  also,  many  questions  in  simple  interest  may  be  read- 
ily and  easily  solved  by  aliquot  parts,  after  the  pupil  has  made 
himself  thoroughly  acquainted  with  the  tables,  which  he  should 
commit  to  memory.     One  number  is  an  aliquot,  or  even  part  of 
another  when  it  forms  an  exact  part  of  it,  as  25c.  is  an  aliquot 
partof  aD=i;  75c. =f;  10c.  =  ^1^;  5c.  =  2V>  ^^*     ^^  ^^'^^  ^^ 
aliquot  part  of  a  cwt.=:^;  28  lbs.=^;  14lbs.=:^;  7  lbs.=:yi^,  ^g. 


PRACTICE. 


137 


Many  questions  that  occur  in  practice  may  be  easily  solved  by 
decimals  ;  as  it  is  more  immediately  connected  with  sterling 
money,  and  when  pounds,  shillings,  and  pence,  occur,  no  method 
of  calculation  is  better  than  the  various  rules  of  practice. 


1 .  When  the  price  is  in  dollars  and  cents,  multiply  the  given 
quantity  by  the  dollars,  and  take  aliquot  parts  for  the  cents,  and 
add  the  products  together,  and  you  will  have  the  answer  in 
D.,  c,  &c. 

2.  But  when  the  quantities  are  of  various  denominations, 
write  down  the  given  price  of  one  of  the  highest  denomination, 
and  multiply  it  by  the  whole  of  the  highest  denomination  ;  then 
take  aliquot  parts  of  the  next  lowest  denomination  continually, 
then  add  the  products  together  for  the  answer. 

Proof,  by  Simple  Proportion. 

A  TABLE  OF  ALIQUOT,  OR  EVEN  PARTS. 


Time. 


6  mo.  is 

4  mo.  is 
3  mo.  is 

2  mo.  is 

1  mo.  is 
equal  to 
15  days 
10  days 
7^  days 
6    days 

5  days 

3  days 

2  days 


i  a 
h  a 
i  a 
i  a 
tV  a 
-i-of 
^of 
lof 
I  of 
I  of 
iof 

1 


of 


year 
year 
year 
year 
year 

3  mo. 

a  mo. 

a  mo. 

a  mo. 

a  mo. 

a  mo. 

a  mo. 

a  mo. 


Cubic  measure. 


96  ft.  is 
64  ft.  is 
32  ft.  is 
16  ft.  is 
8  ft.  is 
4  ft. 
2  ft. 


Dolls. 


50c. 

33ic. 

25c. 

20c. 

12.5c. 

10 

H 

5 

4 
2 
1 


i  of  a  D, 
i  of  a  D. 
:!  of  a  D. 
^  of  a  D. 
1  of  a  D. 
of  aD. 


1 

10 

yV  of  a  D. 

/q  of  a  D. 

^0  of  a  D. 
is  ^  of  a  D. 
is  -J^T  of  a  D. 


Avoir.  Weight. 


84  lb.  is 
56  lb.  is 
28  lb.  is 
14  lb.  is 
7  1b.    is- 


is 


50 

T^of 


D. 


f  of  a  cord 

^  of  a  cord 

i  of  a  cord 

i  of  a  cord 

is  j^g  of  a  cord 

is  ^^  of  a  cord 

is  -^-g  of  a  cord 


Land  Measure. 

80po.or2R.ianA 

40po.orlR.^an  A. 

32  po, 

20  po 

16po 

8po. 

4  po. 

2po. 


12* 


^an  A 
J- an  A. 

yi^anA. 

2V  an  A. 

4V  an  A. 

^anA. 


f  of  a  cwt. 

J  of  a  cwt. 

^  of  a  cwt. 

-|-  of  a  cwt. 
Jg  of  a  cwt. 
3.5  lb.  is  ^^2  of  a  cwt. 
If  lb.  is  gif  of  a  cwt. 

Cloth  Measure. 

27  in.  or  3  qrs.  |-  yd 
18  in.  or  2  qrs.  -i  yd 
9  in.    or  1  qr.     ^  yd 

iVyd 


4.5  in.  or2na. 
2.25  orlna. 
Uin. 


32 


yd 


Parts  of  a  i  cwt. 


14  lb.  is  J  of  a  qr 
7  lb.  is  ^  of  a  qr 
4  lb.  is  ^  of  SL  qr 
2  lb.  is  -^^  of  a  qr 
1  lb.    is  ^  of  a  qr. 


138  PRACTICE. 


EXAMPLES. 


1 .  What  is  the  value  of  5  cwt.  1  qr.  14  lbs.,  at  D2.50  per  cwt.  ? 
Thus  :     J  D2.50  First  multiply  D2.50  by  5  cwt., 

5  X  and  this  will  give  the  value  of  5 

cwt. ;  then  A  of  D2.50  is  62ic., 

12.50=5  cwt.      the  price  or  value  of  the  1  qr. ; 

I         62 J  i  cwt.      then  the  ^  of  62  J  is  31^  the  value 

=31-J-  14  lbs.      of  14  lbs.,  and  their  products  give 

the  whole  value,  &c. 

Dl3.43f 

2.  What  cost  2  cwt.  3  qrs.  7  lbs.  8  oz.  of  iron,  at  5  dollars 
per  cwt.  ? 

J)5.00D.         Or,  1792  oz.  :  5048  oz.  ::  5.00D. 
2  cwt.  5.00 

D,  c.  m. 

10.00  =2  cwt.  1792)2524000(14.08.4+  Arts    proof. 

3.75  =:$  cwt.       ^  ^    *  ^  >^. 


.31^=7 
.02 


s.^f=^>7y  ^^^^^""^^^Hi 


D14.08i  Arts. 

3.  What  is  the  value  of  498  yards  of  tape,  at  6  mills  per 
yard  ?  Ans.  D2.98.8. 

4.  What  cost  1724  yards  of  flannel,  at  37^  cents  per  yard  ? 

Ans,  D646.50. 

5.  What  cost  190  pounds  of  cotton,  at  20  cents  per  pound  ? 

Ans.  D38.00. 

6.  What  cost  16  cwt.  2  qrs.  of  sugar,  at  D5.18  per  cwt.  ? 

Ans.  D85.47. 

7.  What  cost  560  yards  of  sheeting,  at  c.20.5  per  yard  ? 

Aw5.Dll4.80. 

8.  What  cost  270.5  yards,  at  15  cents  per  yard  ? 

Ans.  D40.57.5. 

9.  What  cost  45  gallons,  at  18|  cents  per  gallon  ? 

Ans.  D8.43.75. 

10.  What  is  the  cost  of  5^.5  pounds  of  butter,  at  c.12.5  per 
pound  ?  Ans.  D7.06.25. 

11.  What  is  the  cost  of  15  A.  1  R.  at  D  10.25  per  acre  ? 

Ans.  D156.31.25 

12.  What  cost  5.30  cords  of  wood,  at  D2.50  per  cord  ? 

Ans.  D13.25. 

13.  What  cost  6  yards,  3  quarters  of  cloth,  at  D1.50  per  yard.? 

Ans.  010.13^, 


PRACTICE.      -^^^^^  I%^X 

14.  What  cost  30.5  bushels  of  oats,  at  37|  cts.  per  bushel  ? 

Ans.  Dll.43.75. 

15.  What  cost  8  bushels,  3  pecks  of  wheat,  at  Dl.12  per 
bushel?  Ans.  D9.80. 

16.  What  cost  18.25  bushels  of  rye,  at  93.75  cts.  per  bushel  ? 

Ans.  D17.11. 

17.  What  cost  25  bushels  of  com,  at  56.25  cts.  per  bushel  ? 

Ans.  D14.06.25. 

18.  What  cost  2.75  bushels  of  cloverseed,  at   D4.25  per 
bushel?  Ans.  Dl  1.68.75 

19.  What  cost  12  cwt.,  2  qrs.,  14  lbs.  of  rice,  at  D4.75  per 
cwt.  ?  Ans.  D59.96.8f, 

20.  What  is  the  value  of  130  cwt,  1  qr.,  at  Dl5  per  cwt.  ? 

Ans.  D1953.75. 

21.  What  is  the  value  of  25  cwt.  1  qr.  9  lbs.,  at  Dl.75  per 
cwt.?  Ans.  D44.32.84- 

22.  What  is  the  value  of  6  lbs.  5  oz.  10  pwt.  5  gr.,  at  D4.16 
per  lb.  ?  Ans.  D26.86.9. 

23.  What  is  the  value  of  428  gallons,  3  quarts,  at  Dl.40  per 
gallon?  Ans.  D600.25 

24.  What  is  the  value  of  5  hogsheads,  31^  gallons,  at  D47.1Sj 
per  hogshead  ?  Ans.  D259.16 

25.  What  is  the  value  of  35  A.  2  R.  18  po.,  at  D54.35  pei 
acre?  Ans.  D1935.53.9 

26.  What  is  the  value  of  750  A.  1  R.  4  po.,  at  D12.25  pei 
acre?  Ans.  D9190.86.8+ 

27.  What  cost  927  yards,  at  53j  cents  per  yard? 

Ans.  D494.40 

28.  What  cost  2691  lbs.,  at  16f  cents  per  lb.  ?        D44.91.7, 

29.  What  cost  300^  lbs.,  at  171-  cents  per  lb.  ?     D52.58.7^. 

30.  What  cost  265  lbs.,  at  12^  cents  per  lb.  ?  (-^8)z= 

Ans.  D33.12.5. 

31.  What  cost  1000  yards,  at  12.5  cents  per  yard  ?       D125 

32.  If  1000  yards  cost  D1625,  what  cost  1  yard  ?    Dl.62.5 

33.  What  cost  862^  feet  of  boards,  at  at  D12.00  per  M.  ? 

Ans.  DIO.34.6 

34.  What  cost  35^  feet,  at  35  cents  per  foot  ?  D12.32 

35.  What  cost  842  yards,  at  66f  cents  per  yard  ?  D561.33i. 

36.  What  cost  4  A.  7  R.  20  po.,  at  D25  per  acre  ?  D146.87^. 

37.  What  cost  164  A.  2  R.  15  po.,  at  D75  per  acre  ? 

Ans.  D12344.53.125. 

38.  What  cost  94  yards,  3  qrs.  2  na.,  at  D5.50  per  yard  ? 

Ans.  D521.81.25. 

39.  What  cost  30  bushels,  1  peck,  7  quarts  of  oats,  at  50 
cents  per  bushel  ?  Ans.  D15.23.4 


140  SIMPLE    INTEREST. 

40.  What  cost  7  cwt.  2  qrs.  9  lbs.  12  oz.,  at  D2.50  per  cwt.  ? 

REVIEW. 

What  is  Practice  ?  What  is  an  aliquot  part  ?  Repeat  the 
table  of  aliquot  parts  ?  In  what  cases  will  practice  apply  ? 
Repeat  the  rule. 


SIMPLE  INTEREST. 


Interest  is  an  allowance,  or  premium,  paid  by  the  borrower 
to  the  lender,  or  the  debtor  to  the  creditor,  on  money, notes,  bonds, 
mortgages,  &;c.,  which  is  generally  established  by  law  at  a  cer- 
tain rate,  per  centum,  per  annum  ;  which  means,  so  many  cents 
for  the  use  of  one  dollar,  or  so  many  dollars  for  the  use  of  one 
hundred  dollars,  according  to  the  laws  of  the  state,  or  the  terms 
agreed  upon  between  the  parties,  which,  in  most  cases  in  this 
country  is  6  per  cent.,  or  6  dollars  for  the  use  of  100  dollars,  for 
one  year,  and  in  the  same  proportion  for  a  larger  or  less  sum, 
and  for  a  longer  or  shorter  period.  In  England  the  rate  is  five 
per  cent.  In  the  New  England  states  it  is  six,  and  in  the  state 
of  New  York  it  is  seven  per  cent.  In  all  transactions  with  the 
general  government,  interest  is  computed  at  six  per  cent.  The 
courts  of  the  United  States  allow  interest  according  to  the 
practice  of  the  state  where  the  suit  was  commenced.  The 
banks  throughout  the  Union  compute  interest  at  six  per  cent., 
allowing  the  year  to  consist  of  360  days,  or  30  days  to  the 
month.  When  interest  is  regulated  by  a  statute  of  the  state  or 
country,  at  a  certain  rate  per  cent.,  to  extort,  or  receive  inten- 
tionally, more  interest  than  is  allowed  by  the  statute,  is  called 
usury,  and  is  a  punishable  offence.  This,  however,  is  not  intend- 
ed to  apply  where  the  method  of  calculation  is  erroneous,  but 
where  it  is  evident  fraud,  or  extortion  is  actually  intended. 
Interest  is  of  two  kinds,  namely,  simple  and  compound. 
Simple  interest  is  that  which  is  allowed  on  the  principal  only. 
Commission,  brokerage,  insurance,  &c.,  anything  rated  at  so 
much  per  cent.,  may  be  calculated  by  the  rules  of  interest.  In 
computing  interest,  there  are  four  parts  which  should  be  partic- 
ularly attended  to,  namely  : 

1 .  Principal  is  the  sum  for  which  interest  is  to  be  computed. 

2.  Ratio,  or  rate  per  cent,  per  annum,  is  the  interest  of  one 
nundred  dollars  for  one  year. 


SIMPLE    INTEREST.  I^' 

3.  Time  is  the  number  of  years,  months,  days,  &c.,  for  which 
interest  is  to  be  computed. 

4.  Amount  is  the  principal  and  interest  added  together. 

To  compute  interest  for  days, 

RULE    I. 

Multiply  the  principal  by  the  number  of  days  ;  that  product 
by  the  rate  per  cent.,  and  divide  by  365,  and  the  quotient  will 
be  the  answer  in  dollars,  cents,  and  mills. 

QUESTIONS. 

1.  Required  the  interest  of  550  dollars  for  92  days,  at  6  and 
7  per  cent.  1 

First,  thus  :  550  dollars.  Second,  550  dollars. 

92  days.  92  days. 


50600 

50600 

6  ratio. 

7  ratio, 

D. 

D. 

365)303600)8.31. 7+ J.n5.    365)354200(9.70.4+    Ans. 

Or,    100  dollars  550  dollars  )    a     .-  oj 

365  days.         •'         92  days.  '''      I    ^  ^^^^^-      ^^^ 


>    7  ratio.     4th. 


100  dollars.      ,       550  dollars. 
365  days.         *         92  days. 
2.  Required  the  interest  of  700  dollars  for  365  days,  at  6  per 
cent,  per  annum. 

365  days.  700  dollars. 

700  dollars.  6  ratio. 


255500  D42.00  proof. 

6 
D. 


365)1533000(42.00  Ans. 

3.  Required  the  interest  of  D150  for  63  days,  at  6  per  cent 

Ans.  D1.55.3  + 

4.  What  is  the  interest  of  D250  for  60  days,  at  4.5  per  cent.  ? 

Ans.  Dl.84.9-j- 

5.  What  is  the  interest  of  D  104.25  for  90  days,  at  5  per  cent. 

Ans.  Dl.28.5 

6.  What  is  the  interest  of  D85  for  30  days,  at  7  per  cent.  ? 

Ans.  c,48.9 


142  SIMPLE    INTEREST. 

7.  What  is  the  interest  of  D40  for  50  days,  at  9  per  cent.  ? 

Ans,  0.49.3 

8.  Required  the  interest  on  a  bond  for  D45,  for  150  days,  at 
5  per  cent.  ^  Ans.  c.92.4. 

9.  Required  the  interest  of  D300  for  180  days,  at  7  per  cent. 

Ans.  DlO.35.6. 

10.  Required  the  interest  of  D34.2.76  for  176  days,  at  7i  per 
cent.  Ans.  D 11. 98.2  + 

11.  What  is  the  interest  of  D989.94  for  489  days,  at  5^  per 
cent.  ?  Ans.  D72.94.3  + 

12.  What  is  the  amount  of  D289.64  for  579  days„at  16i  per 
cent.  ?  Ans.  D365.45. 

Interest  for  months^  weeks,  <SfC, 


1.  At  6  per  cent.,  each  dollar  will  draw  1  cent  a  month,  there- 
fore multiply  by  half  the  number  of  months  ;  the  product  will  be 
in  cents,  which  is  the  answer. 

2.  For  the  interest  at  any  other  rate  per  cent.,  take  aliquot 
parts  of  the  interest  at  6  per  cent.,  and  add  it  or  subtract  it  as 
the  case  requires. 

3.  Or  you  can  state  the  questioil  in  compound  proportion. 
13.  Required  the  interest  of  DlOO  for  10  months,  at  6  and  7 

per  cent. 

100  D. 

5  half  no.  mo. 


|-)5.00  at  6  per  cent. 
+83.3 


D5.83.3  at  7  per  cent. 
14.  What  is  the  interest  of  D94.50  for  19  months,  at  9  per  ct. 

Thus :     100  D.       :       94.50  J).     ::     )      ^  ^  ,. 
TO  in  ^9  ratio. 

12  mo.  19  ) 


Or,     94.50  D. 

9.5  half  no.  mo. 


for  3^)8.97.75=6  per  cent. 
4.48.87 


D13.46.625  Ans.  at  9  per  cent. 


SIMPLE    INTEREST.  14S 


15.  What  is  the  interest  of  D649.76  for  74  months  at  6  per^ 
cent  "   X37.  Ans.  D240.41.1. 

16.  What  is  the  interest  of  D331.42  for  150  weeks  at  7  per 
cent.?  DlOO  331.42 

W.  52      •  150 


I  7  :  D66.92.1  Ans. 


17.  What  is  the  interest  of  D284.10  for  64  weeks  at  3^  per 
tent.?  Ans.  D12.23.8 

18.  What  is  the  interest  of  D80  for  9  months  at  6  per  cent. 

Ans.  D3.60. 

19.  What  is  the  interest  of  D94  for  13  months  at  6  per  cent.  ? 

Ans.  D6.11. 

20.  What  is  the  interest  of  D75  for  3  months  at  6  per  cent.  ? 

Ans.  Dl.12.5. 

21.  What  is  the  interest  of  D98  for  4  months  at  6  per  ^ent.  ? 

Ans.  Dl.96. 

22.  What  is  the  interest  of  D400  for  9  months  at  6  per  cent.  ? 

Ans.  D18. 

23.  What  is  the  interest  of  D700  for  10  months  at  6  per  cent.  ? 

Ans.  D35. 

24.  What  is  the  interest  of  D80  for  10  months  at  9  per  cent.  ? 

Ans.  D6. 

25.  What  is  the  interest  of  D300  for  15  months  at  6  per  cent.  ? 

Ans.  D22.50. 

26.  What  is  the  interest  of  DlOO  for  11  months  at  5  per  cent.  ? 

Ans   D4.58.4. 
100 
5.5 


i)5.50D.  at  6  per  cent. 
—91.6 


4.58.4D.  Ans. 
Interest  for  years. 

RULE    III. 

Multiply  the  principal  by  the  rate  per  cent.,  and  separate  the 
two  right-hand  figures  for  cents,  and  those  at  the  left  of  the  point 
will  be  dollars,  and  this  will  be  the  answer  for  1  year ;  if  more 
than  one  year  is  required,  multiply  this  interest  by  the  number 
of  years,  and  the  product  will  be  the  answer.  If  there  are  frac- 
tional parts  of  a  year,  you  can  work  by  aliquot  parts,  &c. 


144  SIMPLE    INTEREST. 

27.  What  is  the  interest  of  D500  for  1  year,  for  3  years,  foi 
3^  years,  and  for  4J  years,  at  9  per  cent.  ? 

Ans.  D45  for  1  year,  D135  for  3  years,  D  157.50  for  3^  years, 
D]  91.25  for  4^  years. 

500  500 

9  per  cent.  9 


^)45.00=^1  year.  ^)45.00c=l  year. 

3X  4x 


D135.00  =  3  years.  D180.00=4  years. 

22.50=-|  year.  1 1 .25=^  year. 

D157.50=r3iyrs.  Ans.     0191.25=^4^  years  Ans. 

28.  What  is  the  interest  of  D  150.25  for  1  year  at  6  per  cent. 

Ans.  D9.01.5. 

29.  What  is  the  interest  of  D67for  7  years  at  6  per  cent,  per 
annum?  Ans.  D28.14. 

30.  What  is  the  interest  of  D75  for  10  years  at  6  per  cent.  ? 

Ans.  D45. 

31.  What  is  the  interest  of  D89  for  11  years  at  6  per  cent.  1 

Ans.  D58.74. 

32.  What  is  the  interest  of  D700  for  12  years  at  6  per  cent.  ? 

Ans.  D504. 

33.  What  is  the  interest  of  2000  for  9  years  at  6  per  cent.  ? 

Ans.  D1080. 

34.  What  is  the  interest  of  300  for  14  years  at  6  per  cent.  ? 

Ans.  D252. 

35.  What  is  the  interest  of  D250.25  for  3  years  at  7  per  ct.? 

Ans.  D52.55.25. 

36.  What  is  the  interest  of  D442  for  4  years  at  8  per  cent.  ? 

Ans.  D141.44. 

37.  What  is  the  interest  of  D222  for  6  years  at  5  per  cent.  ? 

Ans.  D66.60. 

38.  What  is  the  interest  of  D250  for  9f  years  at  5j  per  ct.  ? 

Ans.  D134.06J 

Interest  for  years,  months,  weeks,  days,  <SfC, 

RULE    IV. 

Find  the  interest  of  the  given  sum  for  1  year ;  then  as  1  year 
ie  to  the  given  time,  so  is  the  interest  of  the  given  sum  for  1 
year  to  the  interest  required ;  or,  take  part  of  the  yearly  interest 
for  the  aliquot  parts  of  a  year  that  are  in  the  given  time,  and 
add  the  interest  of  the  days  (if  any),  which  find  by  rule  1st. 
When  fractions  occur  in  the  rate  per  cent.,  work  decimally. 


SIMPLE    INTEREST. 


145 


39.  What  is  the  interest  of  D95.25  for  4  years,  1  month,  5 
days,  at  7  per  cent. "? 

Thus:     D95.25 

7  rate  per  cent. 


^)D6.66.75— 1  year, 
4X 


26.67.00—4  years. 
I)     .55.56—1  month, 
9.25 — 5  days. 


da. 
365 


D27.31.81  Ans, 

da.  D. 

Or     365     :     1495     ;;     6.66.75     :     27.31 -f 

40.  What  is  the  interest  of  D350  for  3  years  and  10  months, 
at  6  per  cent,  per  annum  ? 

D350 

6  rate  per  cent. 


J  &^)21.00— 1  year. 
3X 


63.00-— 3  years. 
=  10.50- 
21-^3)=  7.00 


21-^2)3=10.50-^6  months,  } 
4  months.  > 


:10 


D80.50  Ans, 
DlOO         :         D350 


l^  n^onths,         46  mo 


■*    >  6  rate  per  cent, 
no.  )  ^ 


13.00 


16100 
6x 


12)96600 

D80.50  Ans.  D. 

Or,  12  months  :  46  months     :     D21  interest.     80.50. 

41.  Required  the  interest  of  D200  for  15  months  at  6|  per 
cent.  Ans.  200xl5mo.  6^ X rate  per  ct.~12mo.  — D16.25. 

42.  What  is  the  interest  of  D500  for  2  years  and  9  months, 
ftt  6  per  cent.  ?  Aris.  82.50. 

13 


146  SIMPLE    INTEREST. 


Thus,  500 

6  XT.  As  12  mo.  :  33  mo.  ::  30 

ID 

1 

Ist/ 
znt.  :  82.51  < 

D30.00  interest. 

2d.                       3d. 
100             500        >  ^    5      500 
12     •          33  ••    $^    >         16ix 

4th. 
500 
6x 

D82.50 

1)30.00—1  year. 
2x 

60.00 — 2  years 
22.50 — 9  mo. 

D82.50  Ans, 

43.  What  is  the  interest  of  D250  for  1  year  and  9  months 
at  6  per  cent.  ?  Ans.  D26.25. 

44.  Required  the  interest  of  D7500  for  4  months  at  7  per  ct. 

Ans.  D175. 

45.  What  is  the  interest  of  D967  for  2  years  and  4  months 
at  6  per  cent.  ?  Ans.  D135.38. 

46.  What  is  the  interest  of  D1260  for  8  months  at  7  per  ct.? 

Ans.  D58.80. 

47.  What  is  the  interest  of  D450  for  6  months  and  20  days 
at  5.5  per  cent.  ?  Ans.  D13.75. 

48.  Required  the  amount  of  D240  for  61  days  at  4.75  per 
cent.  Ans.  D241.90. 

49.  What  is  the  interest  of  Dl  for  15  months  at  6  per  cent.  ? 

Ans.  DO.07.5. 

50.  Required  the  amount  of  D672.45  for  7  years,  4  months, 
25  days,  at  7\  per  cent. 

51.  Required  the  interest  of  D379.50  for  27  years,  9  months, 
5  days,  at  6.7  per  cent. 

Bank  interest. 

Remarks. — The  method  of  computing  interest  in  the  banks  is 
at  the  rate  of  30  days  to  the  month,  or  360  days  to  the  year  ;  this 
will  make  the  interest  too  much ;  the  difference  may  be  discov- 
ered by  comparing  the  calculations  below  with  rule  1st ;  but  this 
is  not  all ;  it  is  usual  to  take  the  interest  in  advance,  which  is 
sometimes  improperly  called  discount ;  this  error  is  greater  than 
the  first,  for  instead  of  6,  it  is  little  less  than  7  per  cent.  Thus, 
if  A.  should  borrow  DlOO  of  the  bank,  or  give  his  note  for  that 


SIMPLE    INTEREST.  147 

sum,  and  the  bank  should  deduct  the  interest  for  one  year,  which 
would  be  six  dollars,  and  keep  that  interest  for  one  year ;  the 
interest  of  which  would  be  36  cents.  It  would  stand  thus  :  If 
the  bank  will  gain  D6.36  on  D94  loaned  to  A  ,  how  much  will 
DlOO  gain?  As  D94  :  D6.36  ::  DlOO  :  D6.76  57  Ans.  I 
believe  the  above  is  generally  practised  in  all  banks  throughout 
the  United  States,  with  the  addition  of  3  days  (commonly  called 
days  of  grace)  which  are  allowed  the  payer  on  all  notes,  after 
the  time  expires  for  which  they  were  drawn. 

RULE    V. 

Multiply  the  dollars  by  the  number  of  days,  and  divide  the 
product  by  6,  and  the  quotient  will  be  the  answer  in  mills ; 
which  point  off  into  dollars,  &c.  Or,  work  by  rule  1st,  and 
divide  by  360,  the  answer  will  be  the  same. 

52.  What  is  the  interest  of  D1542  for  90  days,  at  6  per  cent.  ? 

Thus:         1542  D.  Or,         1542  D. 

90  days.  90  days. 


6)138780  138780 

6x 

D23.13  Ans.  D. 

360)832680(23.13 

53.  What  is  the  interest  of  D70  for  90  days,  at  6  per  cent.  T 

Ans.  Dl.05. 

54.  What  is  the  interest  of  D98  for  90  days,  at  6  per  cent.  ? 

Ans.  D1.47. 

55.  What  is  the  interest  of  D400  for  90  days,  at  6  per  cent.  ? 

Ans.  D6.00. 

56.  What  is  the  interest  of  D950  for  90  days,  at  6  per  cent.  ? 

Ans.  D14.25. 

57.  What  is  the  interest  of  D499  for  57  days,  at  6  per  cent.  ? 

Ans.  D4.74. 

58.  What  is  the  interest  of  D900  for  59  days,  at  6  per  cent.  ? 

Ans.  D8.85. 

59.  What  is  the  interest  of  D3000  for  60  days,  at  6  per  cent.  ? 

Ans.  D30. 

60.  What  is  the  interest  of  D600  for  103  days,  at  6  per  cent.  ? 

Ans.  D10.30 

61 .  What  is  the  interest  of  D3000  for  104  days,  at  6  per  cent.  ^ 

Ans.  D52.00 


148  SIMPLE    INTEREST. 

Interest  on  notes,  cj-c,  where  partial  payments  have  been  ?nade. 

As  it  respects  the  computation  of  interest  on  notes,  or  bonds, 
where  partial  payments  or  endorsements  have  been  made,  there 
are  various  ways  :  in  some  states  the  rule  is  established  by  law, 
in  others  it  is  not  regulated  by  statute,  but  the  courts  of  equity 
adopt  such  rules  as  they  think  proper. 

The  limits  of  this  work  will  not  permit  a  full  investigation  ; 
several  rules  and  examples  will  be  given,  and  the  principle  left  to 
the  action  of  those  concerned,  as  no  rule  given  by  any  author 
will  be  strictly  adhered  to.  The  following  appears  to  be  a  fair 
and  equitable  method  of  computing  interest,  when  endorsements 
are  made  ;  it  is  always  presumed  that  the  note  is  due  when  pay- 
ments are  made,  and  that  the  holder  of  the  note  is  entitled  to  the 
interest  accruing  thereon,  whether  the  note  calls  for  periodical 
payments  or  not ;  the  intention  is  to  give  the  interest  to  the  per- 
son who  has  a  just  demand,  and  is  entitled  to  receive  the  prin- 
cipal, as  well  as  interest.  Cases,  hovvev^er,  may  occur  where  a 
departure  from  this  rule  may  be  necessary,  as  when  there  are 
endorsements  made  before  the  note  becomes  due  ;  in  this  case 
the  person  making  the  endorsement  is  entitled  to  compute  inter- 
est on  it  to  the  time  the  note  becomes  due,  but  no  longer,  for 
then  the  holder  of  the  note  may  demand  payment  in  full. 

62.  On  demand,  I  promise  to  pay  A.  B.,  or  ®rder,  the  sum  of 
five  hundred  dollars,  with  interest,  for  value  received.  D500.00. 

January  1,  1843.  C.  P. 

Endorsements  :     April  1,  1843,  D200  ;    Oct.  1,  1843,  D200. 

1st.  D500  princ. 

6  per  cent., 


D30.00  int.  1  year. 
Dl2. —  int.  on  endors. 


D18  bal.  int. 
DlOO+Jan  1,  1844. 

D118  amount  due. 


'      In  this  case  C.  P.  has  the  use  of  the  ' 
interest  of  the  endorsement. 
D9— 9mo.  =  40.5  c.   .  . 
3     3"         4.0       ^  ^' 


200  end.  3  months. 
1.5 

D3.00 

interest. 

D200- 
4.5 

-9  mo. 

D9.00 
3.00 

D12.00  interest  on  en- 
dorsement. 

SIMPLE    INTEREST.  14& 

2d.  D500  princ.  April  1,  3  months. 

1.5 

D7.50  interest  for  3  months  paid 

500  princ. 
— 200  endorsement. 

D300  April  1,  balance,  princ. 
3  for  6  months,  1st  October. 


D9.00  interest  paid. 

interest  paid. 

3.00  princ.                           D7.50 

— 2.00  endorsement.                  9.00 

1.50 

DlOO  princ.  balance  for~ 

1.5  mo.  3  mo.  to 

18.00 

Dl.50  int.  p.  Jany.  1. 

> 

DlOO—  1844. 

The  2d  example  is  correct.  A.  B.  had  a  right  to  receive  the 
interest  on  D9  9mo. :  D3  3mo.=44.5,  this  is  the  difference  in 
the  1st  and  2d  examples. 

63.  A.  received  of  B.  his  note  for  DlOO,  dated  January  1, 
1820,  payable  in  1  year,  w^ith  6  per  cent,  interest ;  and  July  1, 
A.  received  of  B.  fifty  dollars,  which  A.  credited  on  the  note, 
what  was  the  balance  due  from  B.  to  A.  on  January  1st,  1821, 
one  year  ? 

Dr.  B.  in  account  with  A.  Cr. 

1821.  1820. 

January  1,  to  his  note  of  DlOO  Julv  1,  by  cash,    D50.00 

1821. 
To  1  year's  interest,  6  Jan.  l,byint.of  D50, 

paid  as  above,         1.50 

D106  By  balance,  54.50 


D106.00 
In  this  case  D54.50  is  the  equitable  balance.  If  the  above 
note  had  been  on  demand,  or  due  at  the  time  of  payment  (July 
1,  1820)  the  balance  on  the  1st  of  January,  1821,  would  have 
been  D54.59,  difference  9  cents,  which  would  have  arisen  on 
the  D3  for  6  months  ;  this  is  the  same  as  the  1st  and  2d  examples. 

13* 


1^0  SIMPLE    I>rrEREST. 

"  We  have  no  special  acts  of  assembly  in  this  sraie.  [Penn.] 
Chief  Justice  McKean,  in  1785,  fixed  the  following  rule  for 
calculating  interest  [see  Dallas,  1st,  p.  124]  :  That  the  inter- 
est of  the  money  paid  in  before  the  time  be  deducted  from  the 
interest  of  the  whole  sum  due  at  the  time  appointed  by  the 
instrument  for  making  the  payment.'*     As  per  examples  above. 

64.  Due  A.  B.  or  order,  the  sum  of  one  thousand  dollars  on 
demand,  with  interest,  for  value  received. 

January  1,  1842.     DlOOO.OO.  C.  D. 

Endorsements  :  July  1, 1842,  D300  ;  January  1,  1843,  D300  ; 
July  J,  1843,  D200  ;  required  the  balance  due  January  1,  1844, 
interest  6  per  cent. 

DIOOO  January  1,  1842. 

3  )  interest  July  1,  1842. 

I  6  months. 

Paid  D30.00 


DIOOO 
endors.— 300 


700  princ.  July  1,  1842. 
3 


paid,  D21.00  interest  July  1,  1843,  , 

6  months.  ; 

princ.  700 
endors.— 300 


400  July  1,  1843, 
3      6  months. 


paid,  D12.00  interest. 

400 
endors.— 200 


200  princ.  January  1,  1844, 
3      6  months. 


interest,  6.00 
200 


D206.00  due  on  the  note. 


SIMPLE    INTEREST. 


151 


The  following  is  the  rule  established  in  the  state  of  New 
York  (see  Johnson's  Chancery  Reports,  vol.  1,  page  17).  The 
same  rule  is  also  adopted  in  Massachusetts  and  in  most  of  the 
other  states. 

RULE    I. 

Compute  the  interest  on  the  principal  to  the  time  of  the  first 
payment,  and  if  the  payment  exceed  this  interest,  add  the  inter- 
est to  the  principal  and  from  the  sum  subtract  the  payment ;  the 
remainder  forms  a  new  principal. 

2.  But  if  the  payment  is  less  than  the  interest,  take  no  notice 
of  it  until  other  payments  are  made,  which  in  all  shall  exceed 
the  interest  computed  to  the  time  of  the  last  payments  ;  then 
add  the  interest  so  computed  to  the  principal,  and  from  the  sum 
subtract  the  sum  of  the  payments ;  the  remainder  will  form  a 
new  principal  on  which  interest  is  to  be  computed  as  before. 

Note, — The  above  rule  does  not  differ  materially  from  the 
preceding  examples,  and  is  the  same  in  its  general  application  ; 
the  object  in  computing  interest  on  notes,  should  be  to  keep 
down  the  interest,  that  it  may  not  form  a  part  of  the  principal 
carrying  interest,  and  in  the  second  place  to  pay  the  interest  to 
the  one  entitled  to  receive  it. 

(When  partial  payments  are  made  at  short  periods,  the  fol- 
lowing rule  will  apply.) 

RULE. 

Subtract  the  several  payments  from  the  original  sum  in  their 
order,  placing  their  dates  in  the  margin. 

65.  Suppose  a  bill  of  D359  was  due  January,  1839,  and  D75 
was  paid  February  3,  D50  March  5,  D80  April  9,  D145  June  7  ; 
what  interest  is  due  at  5,  6,  and  7  per  cent.  ? 
Dates.  Bill. 


January  1 , 
Feb.  3,  paid 

Balance, 
March  5,  paid, 

Balance, 
April  9,  paid, 

Balance, 
June  7,  paid, 


D.  350 
—75 

275 
—50 

225 
—80 

145 
145 


33 

jrruuut 

11550 

30 

8250 

35 

7875 

59 

8555 

36230 

7300)36230(4.963D. 
at  5  per  ct. 


6083)36230(5.95.5 
at  6  per  cent. 


5214)36230(6.94.8 
at  7  per  ct. 
Ans, 


(See  next  page  to  make  a  divisor,  &c.) 
(See  Note  at  the  conclusion  of  Simple  Interest.) 


152  SIMPLE    INTEREST. 

The  amount,  time,  and  rate  per  cent.,  given  to  find  the  prii  cipdl. 

RULE    VI. 

1.  Find  the  amount  of  DlOO  for  the  time  at  the  given  rate 
per  cent. 

2.  Then  as  the  amount  of  DlOO  for  the  time  required,  at  the 
given  rate  per  cent.,  is  to  the  amount  given,  so  is  DlOO  to  the 
principal  required. 

66.  What  principal  at  interest  6  years  at  6  per  cent,  will 
amount  to  D500  ?  thus  D6x6  years=:36D  interest  for  6  years, 
+  100r=:136D.  amount  of  DlOO ;  then  D136  :  D500  ::  DlOO  : 

Ans.  D367.65. 

67.  What  sum  being  put  to  interest  6  years  at  6  per  cent, 
will  amount  to  D44.88  1  Ans.  D33. 

68.  What  sum  being  put  to  interest  8  years  at  6  per  cent, 
will  amount  to  D74  ?  Ans.  D50. 

69.  What  sum  being  put  to  interest  10  years  at  6  per  cent, 
will  amount  to  D 160?  Ans.    DlOO. 

70.  What  sum  being  put  to  interest  11  years  at  6  per  cent, 
will  amount  to  D830  ?  Ans.  D500. 

The  principal,  amount,  and  time,  given  to  find  the  rate  per  cent. 

RULE    VII. 

Find  the  interest  for  the  whole  given  time,  by  subtracting  the 
principal  from  the  amount ;  then  as  the  principal  is  to  DlOO,  so 
is  the  interest  of  the  principal  for  the  given  time  to  the  interest 
of  DlOO,  for  the  same  time  ;  divide  the  interest  last  found  by 
the  time,  and  the  quotient  will  be  the  rate  per  cent. 

71.  At  what  rate  per  ct.  will  D500  amount  to  D605  in  3  years? 


D500 
3x 

605  amount      then  1500  :     105     ::     100 
— 500  principal                               100 

1500 

105  interest,                    1500)10500 

7  per  cent.  Ans, 

72.  At  what  rate  per  cent,  will  D750  amount  to  D930  in  4 
years  ?  Ans.  6  per  cent. 

73.  At  what  rate  per  cent,  will  D200  amount  to  D255  in  5 
years  ?  Ans.  5.5  per  cent. 


I  SIMPLE    INTEREST.  153^ 

I  To  find  the  time  when  the  principal  amount  and  rate  per  cent,  are 

RULE    VIII. 

Divide  the  whole  interest,  by  the  interest  of  the  principal  for 
one  year,  and  the  quotient  will  be  the  time  required. 

74.  In  what  time  will  D400  amount  to  D568,  at  6  per  cent.  1 

D.      y.        D. 
Thus:     D400         568         then  24  :   1   ::   168  :  7  years  ^n^. 
6     —400 


D24.00         168 
75.  In  what  time  will  D600  amount  to  D750  at  5  per  cent.  ? 

Ans.  5  years. 

A  short  method  to  find  the  interest  for  one  or  more  months,  at  any 
rate  per  cent, 

RULE    IX. 

Multiply  the  principal  by  the  rate  per  cent.,  which  will  give 
the  interest  for  1  year ;  divide  this  by  12,  and  the  quotient  will 
be  the  interest  for  1  month  ;  multiply  this  last  interest  by  the 
number  of  months  required,  &:c. 

Example, — D250  princ.  5  per  ct.         D120  at  7  per  ct. 
5X  7X 


12)12.50  for  1  year.  12)8.40  for  1  year. 

D  1.04.2  for  1  month.  0.70  for  1  month. 

7x  8X 


7.29.4  for  7  months.  D5.60  for  8  months. 


then  Dl.04.2  inter,  for  1  month. 

4x  D75  .70c. 

7  per  ct.  11  x  mo. 


D4.16.8for4mo. 


2x  12)5.25.=  1  yr.    D7.70forllmo. 


D8.33.6  for  8  mo.  0.43.7J  for  1  mo. 


154  SIMPLE    INTEREST. 

To  make  a  divisor  for  any  rate  per  cent. 

RULE    X. 

Multiply  365  days  by  DlOO,  and  divide  by  the  rate  per  cent 
•  Thus,  365x100-^6=6083  divisor  at  6  percent. ;  365x100 
-f-5=:7300  divisor  ;  365  X  100-^-7=5214  divisor  at  7  per  cent. ; 
and  so  for  any  other  rate  per  cent. 

76.  Required  the  interest  of  D400  for  26  days  at  5,  6,  and  7 
per  cent. 

Thus,  400  x26=:10400-f-6083=Dl.70.9f  at  6  per  cent.  Ans. 
10400^7300nrDl.42.44.  at  5  per  cent.  Ans.  10400—5214 
=rDl.99.4f  at  7  per  cent.  Ans.     Or  thus,  100       rate   400 

365    :     6  ::  26  • 
Dl.70.9|.     Or  400x26x7-^365  =  Dl.99.4f  at  7  per  cent.; 
Again,  as  365  da.  :  5  per  ct.  ::  7300  :   100  per  ct.,  and  as  12 
mo.  5  per  ct.  ::  240  mo.  :   100  per  ct. 

Hence  it  is  evident,  that  if  the  rate  be  5,  any  principal  will 
give  100  per  cent. ;  that  is,  it  will  double  in  7300  days,  or  240 
months ;  and  at  6  per  cent,  any  sum  will  double  in  6083  days, 
or  200  months  ;  and  at  7  per  ct.  in  5214|^  days,  or  171|-  months  ; 
from  which  the  following  rule  has  its  origin,  as  exemplified 
above : — 

RULE    XI. 

Multiply  the  principal  by  the  days,  and  for  6  per  cent,  divide 
by  6083  ;  for  5  per  cent,  by  7300,  &c.,  and  the  quotient  is  the 
answer.  These  being  the  number  of  days,  in  which  any  sum 
will  double  at  those  respective  rates.  For  months,  multiply  the 
principal  by  the  months,  and  divide  by  200  for  6  per  cent.,  or 
240  for  5  per  cent.,  &c.,  and  the  quotient  is  the  answer. 

77.  Required  the  interest  of  D 150  at  5  and  6  per  cent,  for  30 
months. 

Ansl  D150  X  30-r240=rDl8.75  at  5  per  cent. ;  D150  X  30 - 
200  =  D22.50  at  6  per  cent. 

Hence,  when:  interest  is  to  be  calculated  on  cash  accounts,  or 
accounts  current,  where  partial  payments  are  made,  or  partial 
debts  contracted,  multiply  the  several  balances  into  the  days 
they  are  a*  interest,  which  should  be  done  at  every  transaction, 
and  the  sum  of  tho&e  products  divided  by  6083  and  7300  will 
give  the  interest  at  6  and  5  per  cent.,  and  for  any  other  rate  find 
a  divisor  as  above  directed  (see  quest.  6b). 


SIMPLE    INTEREST.  155 

APPLICATION. 

78.  What  is  the  interest  of  D22  for  25  days,  at  6  per  cent.  ? 
(Rule  1.)  Ans.  c.9. 

79.  What  is  the  interest  of  D58  for  29  days,  at  6  per  cent.  ? 

Ans,  c.27.6. 

80.  What  is  the  interest  of  D400  for  26  days,  at  6  per  cent.  1 

Ans.  D1.71. 
8 1    What  is  the  interest  of  D90  for  28  days,  at  6  per  cent.  ? 

Ans.  c.41.4. 

82.  What  is  the  interest  of  D50  for  9  months,  at  6  per  cent.  1 

Ans.  D2.25. 

83.  What  is  the  interest  of  D57  for  14  months,  at  6  per  cent.  ? 

Ans.  D3.99. 

84.  What  is  the  interest  of  D700  for  1 1  months,  at  6  per  cent.  1 

Ans.  D38.50. 

85.  What  is  the  interest  of  D900  for  1  year  and  1  month,  at 
6  per  cent.  ?  Ans.  D58.50. 

86.  What  is  the  interest  of  D200  for  1  year  and  3  months,  at 
6  per  cent.  ?  Ans.  D15.00. 

87.  What  is  the  interest  of  D500  for  1  year  and  4  months,  at 
6  per  cent.  ?  Ans.  D40.00. 

88.  What  is  the  interest  of  D31  for  10  years  and  6  months, 
at  6  per  cent.  ?  Ans.  D19.53. 

89.  What  is  the  interest  of  D500  for  14  years,  at  6  per  cent.  ? 

Ans.  D420. 

90.  What  is  the  bank  interest  of  D89  for  89  days,  at  6  per 
cent.?  Ans.  Bl. 32, 

91.  What  is  the  interest  of  D764  for  420  days,  at  5.5  per 
cent.?    (Rule  1.) 

92.  What  is  the  interestof  D642.25for  900days,at7.5  per  ct.? 

[n  the  solution  of  the  following  questions,  the  year  will  con- 
sist of  365  days,  and  the  months  of  30^  days.     (Rule  1.) 

93.  What  is  the  interest  of  D1062.80  for  2  months,  at  9  per 
cent.?  Ans.  1062.80x61  x9^365  =  Dl5.98.5. 

94.  What  is  the  interest  of  D1S896.25  for  4  months,  at  4  per 
cent.?  ^71^.  D 172. 42.1  + 

95.  What  is  the  amount  of  D97.21  for  1  year  and  1  month, 
at  7  per  cent.  ?  Ans  D104. 58. 3. 

96.  What  is  the  interest  of  D3 19.25  for  2  years  and  7  months, 
at  7  per  cent.  ?  Ans.  D57.76.6. 

97.  What  is  the  amount  of  D205.10  for   1  year,  4  months, 
and  2  days,  at  7  per  cent.  ?  Ans.  D224.33.4 


156  SIMPLE    INTEREST. 

98.  What  is  the  interest  of  D4008.50  for  1  year  and  9  months 
at  7i  per  cent.  ?  Ans.  D526.73.3 

99.  What  is  the  amount  of  D107.70  for  7  months  and  5  days 
at  7  per  cent.  ?  Ans.  D112.21.3. 

100.  What  is  the  interest  of  D1121.42  for  11  months,  at  5.5 
per  cent.?  Ans.  56.69-f 

101.  What  is  the  amount  of  D1428.50  for  1  year,  5  months, 
and  5  days,  at  5  per  cent.  ?  Ans.  D  1438.72.4. 

102.  What  is  the  interest  of  D 1892. 50  for  1  year,  at  20  per 
cent.  ?  Ans.  Dl892.50x20^D378.50. 

103.  What  is  the  amount  of  D 1050. 25  for  1  year,  at  4  per 
cent.?  Ans.  D1092.26.0. 

104.  What  is  the  interest  of  D742.18  for  120  days,  at  6  per 
cent.?  A71S.  D14:64. 

105.  What  is  the  amount  of  D  19.60  for  1  year  and  10  months, 
U  41  per  cent.  ?  Ans.  D21.21.9. 

Note  1. — For  the  sake  of  greater  exactness  in  calculating 
interest,  should  any  require  it,  the  divisor,  or  number  of  days 
in  a  year  may  be  365i,  but  the  difference  will  be  unimportant ; 
in  this  case  the  hour  the  note  was  dated  and  the  hour  of  pay- 
ment, should  be  considered. 

Required  the  interest  of  D2500  for  the  first  6  months  of  the 
year  1841,  allowing  the  year  to  consist  of  360,  365,  and  365-J- 
days,  at  6  per  cent,  per  annum.  (Rule  1.) 

January,  31+  February,  28  +  March,  31+April,  30  +  May, 
31+June,  30z=181  days;  2500x181=442500x6=2715000 
dividend- 360,  &c.  1st  Ans.  D75.41.6  ;  2d,  D74.38.3  ;  3d, 
D74.33.2.  It  will  be  seen  that  by  omitting  the  5  days  in  the 
year,  the  difference  is  Dl.03.3  too  much. 

Note  2. — The  erroneous  method  of  computing  interest  on 
endorsements,  and  then  deducting  those  amounts  from  the  face 
of  the  note,  except  in  particular  cases,  may  be  seen  in  the  fol- 
lowing statement :  Suppose  I  borrow  DlOO  at  6  per  cent.,  for 
10  years,  and  pay  D6  at  the  end  of  each  year,  what  will  be  due 
at  the  end  of  10  years  ?  The  amount  of  DlOO  is  D160.  But 
the  first  endorsement  of  D6  has  borne  interest  for  9  years,  the 
second  for  8  years,  the  third  for  7  years,  and  so  on  ;  so  that  six 
dollars  have,  in  fact,  been  drawing  interest  forty-five  years,  and 
thus  produced  D  16.20  of  interest.  This  added  to  the  nine  en- 
dorsements of  D6  each,  gives  D70.20  ;  that  is,  while  I  have 
paid  only  the  annual  interest  of  D6,  the  principal  has  actually 
been  reduced  D16.20;  by  paying  D6  annually  for  25  years, 


INSURANCE,   COMMISSION,    AND    BROKERAGE.  15 

and  computing  interest  on  the  several  endorsements  by  this 
method,  the  whole  principal  would  be  paid,  and  the  lender 
would  be  indebted  to  the  borrower  the  sum  of  D2.00,  while  in 
fact,  the  lender  had  received  no  part  of  the  sum  lent.  The 
dillerence  of  the  rules  depends  on  the  principle  assumed  in 
respect  to  the  time  when  the  interest  becomes  due,  and  which 
party  has  a  right  to  receive  it. 

REVIEW. 

What  is  simple  interest  ?  How  many  parts  are  there  which 
require  attention  in  interest  ?  Name  them.  How  do  you  com- 
pute interest  for  days  ?  What  is  the  interest  of  D1799.89  for 
1700  days,  at  9|  per  cent.  ?  How  can  you  calculate  interest 
for  months  at  6  per  cent. "?  How  can  you  calculate  at  any 
other  rate  per  cent.  ?  How  do  you  compute  interest  for  years.? 
When  interest  is  required  for  years,  months,  days,  &c.,  how 
will  you  proceed  ?  What  is  the  interest  of  1)979.99  for  9 
years,  9  months,  9  weeks,  and  9  days,  at  9.9  per  cent.  ?  What 
is  bank  interest  ?  What  is  the  rule  ?  What  can  you  say  of 
the  correctness  of  the  method  of  calculating  bank  interest  ?  When 
the  amount  and  rate  per  cent,  are  given  to  find  the  principal,  by 
what  rule  will  you  work  ?  What  is  the  rule  when  the  principal, 
amount,  and  time,  are  given,  to  find  the  rate  per  cent.  ?  What  is 
the  rule  to  find  the  time,  when  the  principal,  amount,  and  rate 
per  cent.,  are  given  ?  What  can  you  say  in  relation  to  the  compu- 
tation of  interest  on  notes,  &c.,  where  partial  payments  are  made  l 


INSURANCE,  COMMISSION,  AND  BROKERAGE. 

Insurance  is  an  agreement  by  which  an  individual  or  a  com- 
pany agrees  to  exempt  the  owners  of  certain  property  from  loss 
or  hazard.  The  written  agreement  is  called  the  jwlicy.  The 
premium  is  the  amount  paid  by  him  who  owns  the  property  to 
those  who  insure  it,  as  a  compensation  for  their  risk.  It  is  gen- 
erally so  much  per  cent,  on  the  value  of  property  insured.  Com- 
mission is  an  allowance  made  to  a  factor,  or  a  commission  mer- 
chant, for  buying  and  selling.  Brokerage  is  an  allowance  made 
to  dealers  in  money  or  stocks.  The  calculations  may  all  be  made 
according  to  the  rule  of  simple  interest  for  one  year  (See  rule  3. 

14 


158  INSURANCE,    COMMISSION,    AND    BROKERAGE. 

QUESTIONS. 

1.  Required  the  insurance  on  D60,  at  2^  per  cent.  ? 

Arts.  60X2.5=D1.50. 

2.  Required  the  commission  on  D50  at  2^-  per  cent. 

Ans.  D1.25. 

3.  Sold  goods  to  the  amount  of  D200  at  .75  per  cent,  commis- 
sion ;  how  much  was  received  ?  Ans.  Dl.50. 

4.  What  is  the  commission  on  D89  at  3  per  cent.  ?  D2.67. 

5.  What  is  the  insurance  on  D300  at  4.5  per  cent.  ?  D]  3.50. 

6.  What  is  the  insurance  on  D700  at  5.5  per  cent.  ?  D38.50. 

7.  What  is  the  insurance  on  a  ship  and  cargo  valued  at 
D20000  at  61  per  cent.  ?  A7is.  D1300. 

8.  A  house  valued  at  D 10000  which  is  insured  at  1.5  per 
cent.,  required  the  insurance.  Ans.  D150.00. 

9.  B.  collected  a  county  tax  of  D 10000,  for  which  he  re- 
ceived a  commission  of  3^  per  cent.,  what  sum  did  he  receive  ? 

Ans.  D350.00. 

10.  A.  sold  to  B.  goods  to  the  amount  of  .D99,  and  gained 
5.5  per  cent,  on  the  sale  ;  what  was  his  gain?      Ans.  D5.44.5. 

11.  What  sum  must  you  pay  for  the  use  of  D85  at  6^  per 
cent  ?  Ans.  D5.52.5. 

12.  If  I  borrow  D160,  and  agree  to  pay  1.5  per  cent,  for  the 
use  of  it,  how  much  must  I  pay  ?  Ans.  D2.40. 

13.  Sixty-nine  shares  of  bank  stock,  of  which  the  par  value 
is  D125,  is  at  a  discount  of  8  per  cent.  ;  what  is  its  value  ? 

Ans.  D7935. 

14.  What  would  be  the  insurance  on  a  ship  valued  at  D47520, 
at  i  per  cent.  ?  also  at  ^  per  cent.  ?     Ans.  D237.60  ;  D158.40. 

15.  What  would  be  the  insurance  on  a  house  valued  at 
D 14000,  at  1-1^  per  cent.  ?  at  |  per  cent.?  at  J  per  cent.  ?  at  i  per 
cent.  ?  at  1  per  cent.  ?  Ans.  D210  ;  D105  ;  D70  ;  D46.66  ;  D35. 

16.  What  is  the  insurance  on  a  store  and  goods  valued  at 
D15000,  at  l-J-  per  cent.  ?  at  1|  per  cent.  ?  at  2^  per  cent.  ?  at 
3|  per  cent.  ?  at  4^  per  cent.  ?  at  5^  per  cent.  ? 

17.  If  a  policy  be  taken  out  for  D781.25  to  cover  D625,  re- 
quired the  premium  per  cent. 

Thus,  D781.25  :  625  ::  100  :  80  :  then  100— 80= Ans.  20  per  cU 

18.  It  is  required  to  cover  D759,  premium  8  per  cent.  ;  for 
what  sum  must  the  policy  be  taken  ? 

Ans.  1)100-8  =  92  :   100  ::  759  :  825. 

REVIEW. 

What  is  insurance  ?  commission  ?  brokerage  ?  and  policy  ? 


COINS    AND    CURRENCY. 


159 


COINS,  CURRENCY,  &c. 

A  table  showing  the  weight  and  value  of  the  several  coins  of  the 
United  States. 


Gold 


COINS. 

C  Eagle, 
]  Half-Eagle, 
f  Quarter-Eagle, 
f  Dollar, 
I  Half-Dollar, 
Silver  -^  Quarter-Dollar, 
Dime, 
[  Half-Dime, 
Cent, 
Half-Cent, 


Copper 


Pure. 

Standard. 

Value. 

dwt.      gr. 

dwt. 

gr. 

D.    c.     m 

10     7| 
5     3| 

11 

6 

10  00     0 

5 

15 

5  00     0 

2   13J 

2 

19i 

2  50     0 

15   111 

17 

8 

1  00     0 

7  19^f 

8 

16 

0  50     0 

3  20i| 

4 

8 

0  25     0 

1   13i 

1 

17f 

0  10     0 

0  18^»^ 

0 

20A 

0     5     0 

7  00 

0 

0 

0     1     0 

3   12 

0 

0 

0     0     5 

All  Gold  of  equal  fineness  to  be  valued  at  89c.  per  dwt.,  and 
all  Silver  Coins  of  the  same  fineness  at  Dl.ll  per  ounce.  The 
standard  for  Gold  is  1 1  parts  of  fine  and  1  part  of  alloy.  The 
standard  Silver  is  1485  parts  of  pure  silver  to  179  parts  of 
alloy,  which  is  to  be  wholly  of  Copper. 


A  TABLE  OF  GOLD  COINS. 


Names  of  Coins. 

BRAZIL. 

Johannes,  half  in  pro- 
portion    -     -     -     - 

Dobraon      -     -     -     - 

Dobra 

Moidore,  half  in  propor- 
tion      

Crusado      -     -     -     - 

ENGLAND. 

Guinea,  half  in  propor- 
tion     

Sovereign,  half  in  pro- 
portion    -     -     -     - 

Seven-shilling  piece  - 

FRANCE. 

Double  Louis,  coined 
before  1786  -     ■ 


Weight, 
dwt.     gr. 

Grains 

of  pure 

gold. 

Standard 
before  1st 
Aug.1834 
D.    c.  m. 

18 

16 

34 

12 

759 

30  66  6 

18 

6 

401     5 

16  22  2 

6 

22 

152     2 

6  14  9 

14 

15     8 

59  8 

5 

H 

118     7 

4  79  6 

5 

H 

113     1 

4  57 

1 

19 

39     9 

1  60 

10 

11 

224     9 

9  68  7 

Standard 
after  1st 
Aug.1834 
D.  c.  m. 
17  6  8 
32  71  4 
17  30  5 

6  56  0 
63  0 


4  87  5 
1  70  6 


9  69  4 


160 


COINS   AND    CURRENCY. 


TABLE  OF  GOLD  COINS— Continued. 

Names  of  Coins. 

Louis,  coined  bef.  1786 
Double    Louis,    coined 

since  1786  -  - 
Louis,  do.  -  -  - 
Double  Napoleon,  or  40 

francs  -  -  - 
Napoleon,  or  20  francs 

COLOMBIA,    S.   AMERICA. 

Doubloon  -     -     - 

MEXICO. 

Doubloons,  shares  in 
proportion    -     - 

PORTUGAL. 

Dobraon  -  -  - 
Dobra  -  -  -  - 
Johannes  -  -  - 
Moidore  -  -  - 
Piece  of  16  testoons,  or 

1600  rees     -     - 
Old  Crusado,  400  rees 
New  Crusado,  480  rees 
Milree,  coined  1776  - 

SPAIN. 

Quadruple    pistole,    or 

doubloon,1772,  shares 

in  proportion  -  - 
Doubloon,  1801  -  - 
Pistole,  1802  -  -  - 
Coronilla,   gold    dollar, 

or  Vintem,  1801      - 

UNITED    STATES. 

Eaffle,  coined  before  Ju- 
ly 31,  1834  -     -     - 

Eagle,  coined  since  Ju- 
ly 31,  1834  -     -     - 

The  above  tables  are  calculated  and  arranged  to  the  actual 
use  at  the  United  States  mint,  for  estimating  the  value  of  gold 

Note. — In  England,  where  accounts  are  kept  in  pounds,  shil 
lings,  pence,  and  farthings,  the  pound  is  always  estimated  at  20 


Weight. 
5   5^ 

Grains 
of  pure 

gold. 
112  4 

Standard 

before  1st 

Aug.1834 

4  54  1 

9  20 

4  22 

212 
106 

6 
3 

8  59 
4  29  5 

8   7 
4  ^ 

179 
89 

7 

7  23  2 
3  62  4 

17   8^ 

360 

5 

14  56 

17   8i 

360 

5 

14  56 

34  12 

18   6 

18   0 

6  22 

759 
401 

152 

5 

2 

30  QQ   6 

16  22  2 

16  00  0 

6  14  9 

2   6 
15  15 

19| 

49 
13 
14 
18 

3 
6 
8 
1 

1  99  2 
0  54  9 
0  59  8 
0  73  2 

17   81 
17   9 
4   81 

437 

360 

80 

2 
5 

1 

15  03  0 

14  56  0 

3  64  0 

1   3 

22 

8 

0  92  1 

11   6 

247 

5 

10  00  0 

10  18 

232 

0 

CO/NS    AND    CURRENCY 


IM 


Btiillings,  the  shilling  at  12  pence,  and  a  penny  at  4  farthings. 
The  pound  sterling  is  not  a  coin,  although  there  are  bank  notes 
of  that  denomination.  The  pound  sterling  in  this  country  is 
now  estimated  at  D4. 87. 5,  which  wbuld  make  the  shilling=24-J 
cents  nearly. 

REMARKS. 

The  value  of  the  Eagle  coined  prior  to  the  31st  of  July,  1834, 
is  DlO. 66. 8-f-,  which  contains  247.5  grains  of  pure  gold-|-22.5 
grains  alloy =270  grains,  standard  weight.  By  computation,  it 
will  be  found  that  every  23.2  grains  of  pure  gold  are  equal  in 
value  to  Dl,  and  25.8  grains  of  United  States  standard  are  also 
equal  to  Dl. 

To  find  the  value  of  any  Gold  Coin  whatsoever, 

RULE. 

Reduce  its  weight  of  pure  gold  to  grains  (troy),  and  divide  by 
23.2  for  the  value  in  dollars,  cents,  and  mills. 

1.  Required  the  value  of  lib.  of  pure  gold. 

Thus,  llb.Xl2x20x24-f-23.2  =  D248.27.5ff  Ans. 

2.  Required  the  value  of  lib.  standard  gold  of  the  U.  States. 

Thus,  llb.Xl2x20x24^25.8=D223.25.5|f  Ans, 
A  Table  of  Foreign  Coins,   &c. — Showing  their  value  in 
the  United  States  as  established  by  acts  of  Congress — not 
given  in  the  preceding  tables  : — 


I  Silver. 

[  English  or  French  Crown         -     ,   - 
]  dollar  of  Spain,  Sweden,  and  Denmark 
Dollar  of  Mexico  and  S.  A.  S. 
Five-franc  Piece      -         -         -         - 

Franc     

Pistareen 

Pound  of  Ireland  -  -  -  - 
Pagoda  of  India  -  -  -  - 
Tale  of  China  -         -         -         - 

Millrea  of  Portugal  -  -  -  - 
Ruble  of  Russia  -  -  -  - 
Rupee  of  Bengal  -  -  -  - 
Guilder  of  the  United  Netherlands  - 
Mark-Banco  of  Hamburg 
Livr6  of  Francois  .  -  -  - 
Gold  Ducat  of  Russia      -         -         - 


D 

c. 

m. 

1 

10 

0 

1 

00 

0 

1 

00 

0 

0 

93 

6 

0 

18 

8 

0 

20 

0 

4 

10 

0 

1 

94 

0 

1 

48 

0 

1 

28 

0 

0 

66 

0 

0 

55 

5 

0 

39 

0 

0 

35 

5 

0 

18 

5 

2 

00 

0 

3.  In  50  pounds  sterling,  how  many  dollars  ? 

Ans.  4.87.5x50=243.750 
14* 


162  COINS    AND    CURRENCY. 

4    In  75  tales  of  China,  how  much  United  States  money  ? 

Ans.  Dlll.OO 

5.  In  25  sovereigns,  how  many  dollars?        Ans,  D121.87.5 

6.  How  much  United  States  money  in  500  francs  ? 

Ans.  D94.00 


DISCOUNT. 


Discount  is  an  allowance  made  for  the  payment  of  any  sum 
of  money  before  it  becomes  due,  and  is  the  difference  between 
that  sum,  due  some  time  hence,  and  its  present  worth. 

The  present  worth  of  any  sum  or  debt  not  yet  due,  is  so  much 
as  would,  if  put  to  interest,  produce  a  sum  equal  to  the  discount ; 
or  the  interest  of  the  present  worth  and  interest  of  the  discount 
for  the  given  time  and  rate  per  cent,  shall  be  equal  to  the  inter- 
est of  the  given  sum,  or  debt,  for  the  same  time  and  rate  per 
cent.  Thus  the  present  worth  and  discount  of  D 100  for  1  year 
at  6  per  cent. ;  the  present  worth  is  D94.34,  which  subtract  from 
DlOO,  gives  D5. 66  discount,  and  the  interest  of  D94.34  is  D5.66, 
so  that  neither  party  is  wronged,  provided  they  are  both  agreed  ; 
but  in  no  case  should  the  interest  be  allowed  on  the  given  sum 
as  discount,  because  the  interest  would  be  D6.00,  which  is  34 
cents  too  much.  Again,  ijf  I  give  my  note  for  1)106,  payable 
one  year  hence,  th-e  present  value  of  the  note  will  be  less  than 
D106  by  the  interest  on  its  present  value  for  one  year ;  that  is, 
its  present  value  will  be  DlOO.  The  amount  named  in  a  note 
is  called  the  face  of  the  note;  thus  D106  is  the  face  of  the 
above  note ;  the  discount  is  the  difference  between  the  face  of 
a  note  and  its  present  value — that  is,  D6  is  the  discount  on  the 
above  note. 


EXTRACT. 

That  an  allowance  ought  to  be  made  for  paying  money  before 
it  becomes  due,  which  is  supposed  to  bear  no  interest  till  after 
it  is  due,  is  very  just  and  reasonable  ;  but  if  I  pay  it  before  it  is 
due,  I  give  that  benefit  to  another.  Therefore,  we  have  only  to 
inquire  what  discount  ought  to  be  allowed.  Many  suppose  that 
by  not  paying  till  it  becomes  due  they  may  employ  it  at  interest ; 
therefore,  by  paying  it  before  due,  they  shall  lose  that  interest. 


DISCOUNT.  163 

and  for  that  reason  all  such  interest  ought  to  be  discounted  ;  but 
the  supposition  is  false,  for  they  can  not  be  said  to  lose  that  in- 
terest till  the  time  arrives  when  the  debt  becomes  due  ;  in  other 
words,  they  can  not  lose  what  they  do  not  possess,  whereas  we 
are  to  consider  what  would  be  lost  at  present  by  paying  the  debt 
before  it  becomes  due  ;  this  can  in  point  of  equity  be  no  other 
.,han  such  a  sum,  which  being  put  out  at  interest  till  the  debt 
shiil  become  due,  would  amount  to  the  interest  of  the  debt  for 
the  same  time,  as  before  observed. 

The  truth  of  the  rule  for  working  is  evident  from  the  nature 
of  simple  interest ;  for  since  the  debt,  or  face  of  the  note,  may 
be  considered  as  the  amount  of  some  principal  (called  here  pres- 
ent worth),  at  a  certain  rate  per  cent.,  and  for  the  given  time,  that 
amount  must  be  in  the  same  proportion,  either  to  its  principal  or 
interest,  as  the  amount  of  any  other  sum,  at  the  same  rate,  and 
for  the  same  time,  to  its  principal  or  interest.  In  what  is  termed 
hank  discount,  the  interest  is  taken  for,  or  called  discount.  The 
word  is  misapplied  ;  the  banks  loan  money  and  receive  interest, 
not  discount.  The  difference  will  stand  D5. 66  to  D6.76i  on 
DlOO  for  1  year,  at  6  per  cent.     (See  Bank  Interest.) 


I.  Find  the  interest  of  DlOO  for  the  time,  and  at  the  rate  per 
cent,  mentioned  in  the  question ;  then  add  this  interest  to  the 
DlOO,  and  this  is  the  first  term.  2d.  The  given  sum  in  the 
question  is  the  2d  term,  and  DlOO  is  the  3d  term.  3d.  Multi- 
ply the  2d  and  3d  terms  together,  and  divide  by  the  first,  and 
the  quotient  will  be  the  present  worth.  4th.  When  the  discount 
is  required,  subtract  the  present  worth  thus  found,  from  the 
given  sum,  and  the  remainder  is  the  discount. 

QUESTIONS. 

1 .  Required  the  present  worth  and  discount  of  D500,  due  I 
year  hence,  at  6  per  cent.  ? 
Thus:  DlOO 

6  rate  per  ct.,  then  106  :  500  ::  100 

100 

interest — 6.00  Ans, 

100  106)500(471.98.8  pr't  worth, 

D106 

500  given  sum. 
Or,  106)500(471.69.8  present  worth. 


D28.30.2  discount.  Ans. 


164 


DISCOUNT. 


RULE    II. 

Assume  any  principal  at  pleasure,  and  find  the  amount  for  the 
time  and  rate  per  cent.  Then,  as  the  amount  found  is  to  the 
amount  or  debt  given,  so  is  the  principal  assumed  to  the  required 
principal,  or  present  worth. 

2.  Suppose  a  debt  of  D810,  were  to  be  paid  three  months 
hence,  allowing  5  per  cent.,  what  is  its  worth  in  cash  1 


Assume  D80 
5 


then,  81 


3  months  i)4.00 

1 
+  80 


810 
80 


80 


81)64800(800  Ans. 


D81 


Or, 


DlOO     then     101.25  :  810  ::   100 
5  100 


101.25)81000(800  An^. 


i)5.00 

1.25 
100 

D101.25 

3.  Purchased  goods  to  the  amount  of  D750,  on  a  credit  of 
9  months,  at  5  per  cent.,  but  wishing  to  make  immediate  pay- 
ment, it  is  required  to  know  what  sum  in  ready  money  would 
discharge  the  debt.  Ans.  D722.89.1  + 

4.  What  is  the  discount  on  D280,  due  in  6  months,  at  7  per 
cent.  ?  Ans.  D9.46.9-f 

5.  What  is  the  present  worth  of  D840,  due  in  1  year,  6 
months,  at  6  per  cent.  ?  Ans.  D770.64.2. 

6.  What  is  the  present  worth  of  D954,  due  in  3  years,  at  4.5 
per  cent.  ?  Ans.  D840.52.8. 

7.  If  you  purchased  goods  to  the  amount  of  D796.49,  on  a 
credit  of  4  months,  at  3  per  cent.,  what  sum  in  ready  money 
would  discharge  the  debt  ?  Ans.  D788.60.3. 

8.  What  difference  is  therebetween  the  interest  of  D1200  at 
5  per  cent,  per  annum,  for  12  years,  and  the  discount  of  the 
same  sum  at  the  same  rate  and  time  ?  Ans.  D270. 


DISCOUNT.  165 

9.  What  sum  in  ready  money  must  be  received  for  a  bill  of 
900D.  due  73  days  hence,  discount  at  6  per  cent,  per  annum  ? 

Ans.  D889.32.8. 

10.  What  sum  will  discharge  a  debt  of  D615.75,  due  in  7 
months  at  4J  per  cent,  per  annum  ?  Ans.  D600. 

11.  B.  has  D2000  due  him  from  A.,  of  which  D500  are  pay- 
able in  6  months,  D800  in  1  year,  and  the  remainder  at  the  ex- 
piration of  3  years  at  6  per  cent. ;  but  if  A.  should  make  present 
payment,  how  much  would  he  have  to  pay  ? 

Ans.  D1833.37.4+ 
Note. — When  payments  are  to  be  made  at  different  times,  find 
the  present  value  of  the  several  sums  separately,  and  their  sum 
will  be  the  present  value  of  the  note  or  debt. 

12.  What  sum  will  discharge  a  debt  of  DlOOO,  whereof 
D600  is  payable  in  one  year,  and  the  remainder  in  6  months,  at 
4  per  cent.  ?  Ans.  D969.08. 

13.  What  is  the  present  worth  of  D600  due  in  5  years  at  7 
per  cent.  ?  Ans.  D444. 44.44- 

14.  What  is  the  difference  between  the  interest  and  discount 
of  DlOOO  for  1  year  at  6  per  cent.  ?  Ans.  D3.39.6-f- 

15.  What  is  the  present  value  of  a  note  for  D3500,  on  which 
D300  are  to  be  paid  in  6  months,  D900  in  1  year,  D1300  in  18 
months,  and  the  remainder  at  the  expiration  of  2  years,  the  rate 
of  interest  being  at  6  per  cent,  per  annum  ? 

Ans.  D3225.83-f 

16.  What  is  the  present  value  of  D2880,  one  half  payable  in 
3  months,  one  third  in  6  months,  and  the  remainder  in  9  months, 
at  6  per  cent,  per  annum  ?  Ans.  D2810.08-|- 

17.  Bought  goods  to  the  amount  of  D1854  for  which  I  gavo 
my  note  for  8  months  at  6  per  cent.  ;  but  being  desirous  of  tak- 
ing it  up  at  the  expiration  of  2  months,  what  sum  does  justice 
require  me  to  pay  ?  Ans.  D1800 

18.  What  is  the  present  worth  of  D515  due  6  months  hence  ? 
1)500— due  1  year  hence?  D485.84.9 — due  15  months  hence  ? 
D479.06.9— due  20  months  hence?  D468.18.1— due  4  years 
hence  ?  D415.32.2~at  6  per  cent.  ?  Ans.  D2348.42.1-f 

19.  What  is  the  present  worth  of  D1350,  due  5  years  10 
months  hence,  at  6  per  cent.  ?  Ans.  DlOOO. 

20.  What  is  the  discount  of  D460,  due  2  years  6  month-s 
hence  at  6  per  cent.  1  Ans.  D60. 

REVIEW. 

What  is  discount  ?  What  is  present  worth  ?  How  will  you 
first  proceed  to  find  the  present  worth  ?     After  having  found  the 


166  EQUATION. 

interest  of  DlOO  at  the  given  time  and  rate  per  cent,  what  is 
next  to  be  done  ?  After  having  added  the  interest  so  found  to 
DlOO,  by  what  rule  do  you  work  to  find  the  discount  ?  Repeat 
the  rule.  Is  it  correct  to  take  the  interest  for  the  discount  1 
What  is  the  difference  ?  What  is  the  face  of  a  note  ?  When 
payments  are  to  be  made  at  different  times,  how  do  find  the 
present  value  ? 

21.  What  is  the  difference  between  the  discount  of  D227.66 
for  2  years  3  months  and  20  days,  and  the  interest  of  the  same 
sum  for  the  same  time,  at  6  per  cent.  ?  Ans.  D3.81.2.^yfc. 

22.  What  is  the  present  worth  of  D1500  for  90  days  at  7  per     ^ 
cent,  per  annum  ?  ^n^.  D1474.54+  9 

RULE  III.  •  /■  [\}i^ 

1.  Divide  the  given  sum  or  debt  by  the  amount  of  Dl  for  the 
given  time.  2.  The  quotient  will  represent  the  present  worth, 
which  taken  from  the  debt  will  le-ave  the  discount.  Thus  to  find 
the  present  worth  of  D  133.20  payable  1  year  and  10  months 
hence,  and  discount.  The  amount  of  Dl  for  1  year  10  months, 
is  Dl.ll  ;  then  D133.20-f-Dl.ll=Dl20  present  worth;  and 
D133.20— D120=D13.20  discount. 


EQUATION. 

Equation  is  a  rule  used  to  find  the  mean  or  equated  time  of 
several  payments  which  are  due  at  different  times,  so  that  no 
loss  shall  be  sustained  by  either  party. 

RULE  I. 

Multiply  each  payment  by  its  time,  and  divide  the  sum  of  the 
several  products  by  the  whole  debt,  and  the  quotient  will  be  the 
equated  time  for  the  payment  of  the  whole. 

QUESTIONS. 

1.  A.  owes  B.  D380  to  be  paid  as  follows,  namely  :  DlOO  in 
6  months  ;  D 120  in  7  months  ;  and  D160  in  10  months  ;  what 
is  the  equated  time  for  the  payment  of  the  whole  debt  ? 
Thus:     100x6  =  600 
120x7  =  840 
160x10=1600 


380  )3040(8  mo.  Ans, 

2.  I  have  D200  due  me,  of  which  DlOO  is  to  be  paid  in  6 
months,  and  DlOO  in  12  months,  but  it  is  agreed  to  make  one 
TMiyment ;  required  the  time.  Ans,  9  months. 


EQUATION.  117 

3.  A  gentleman  has  due  him  D600  to  be  paid  as  follows,  D400 
in  10  months,  and  D200  in  6  months  ;  what  is  the  equated  time  1 

Ans,  8|  mpnths. 

4.  A.  has  due  him  a  certain  sum  of  money  to  be  paid,  ^  in  2 
months,  5  in  3  months,  and  the  remainder  in  6  months  ;  what  is 
the  equated  time  "?  Ans.  4^  months. 

5.  What  is  the  equated  time  for  paying  D2000,  of  which  D500 
is  due  in  3  months,  D360  in  5  months,  D600  in  8  months,  and 
the  balance  in  9  months  ?  Ans.  6^  months  (nearly). 

6.  What  is  the  equated  time  for  paying  D380  :  whereof  DlOO 
is  payable  in  180  days,  D120  in  210  days,  D160  in  300  days  1 

Thus,  100x180=18000  :  120x210^:25200  :  160x300  = 
48000=91200  dividend -^3 80 =240  days,  An5. 


See  by  rule  1,  at  what  time  the  first  man  mentioned,  ought  to 
pay  in  his  whole  money  ;  then,  as  his  money  is  to  his  time,  so  is 
the  other's  money  to  his  time  ;  inversely,  which,  when  found, 
must  be  added  to,  or  subtracted  from,  the  time  at  which  the  sec- 
ond ought  to  have  paid  in  his  money,  as  the  case  may  require, 
and  the  sum,  or  remainder,  will  be  the  true  time  of  the  second 
payment. 

7.  P.  is  indebted  to  Q.  D150,  to  be  paid,  D50  at  4  months, 
and  DlOO  at  8  months  ;  Q.  owes  P.  D250,  to  be  paid  at  10 
i^onths  ;  it  is  agreed  between  them  that  P.  make  present  pay- 
ment of  his  whole  debt,  and  that  Q.  shall  pay  his  so  much  soon- 
er, as  to  balance  that  favor ;  I  demand  the  time  at  which  Q. 
must  pay  the  D250. 

Thus,  50x4=200;  100x8  =  800  =  1000^  150=6f  months, 
P.'s  equated  time  ;  then,  D150  :  6f  months  ::  D250  :  4  months ; 
then  10  months— 4=6  months,  time  of  Q.'s  payment.  Ans. 

Note. — Notwithstanding  the  general  use  of  the  rules  of  equa- 
tion, they  are  manifestly  incorrect.  It  is  argued  by  those  who 
defend  the  principle,  that  what  is  gained  by  keeping  some  of  the 
debts  after  they  are  due,  is  lost  by  paying  others  before  they  are 
due  ;  but  this  can  not  be  the  case,  for  though  by  keeping  a  debt 
after  it  is  due,  there  is  gained  the  interest  of  it  for  that  time,  yet 
by  paying  a  debt  before  it  is  due,  the  payer  does  not  lose  the  in- 
terest for  that  time,  but  the  discount  only,  which  is  less  than  the 
interest ;  consequently,  the  rule  can  not  be  strictly  correct ;  al- 
though in  most  questions  which  occur  in  business  the  error  is  so 
trifling  that  it  will  always  be  made  use  of  as  the  most  eligible 
method.     The  same  system  of  erroneous  calculation  in  regard 


166  BARTER. 

to  interest  and  discount  is  exhibited  in  equation,  by  a  misappli- 
cation of  terms,  and  a  false  principle. 

Note  2. — Suppose  a  sum  of  money  be  due  immediately,  and 
another  at  the  expiration  of  a  certain  given  time  forward,  and  it 
is  proposed  to  find  a  time,  so  that  neither  party  shall  sustain  loss  ; 
now,  it  is  plain  that  the  equated  time  must  fall  between  the  two 
payments,  and  that  what  is  gained  by  keeping  the  first  debt  after 
''  is  due,  should  be  equal  to  what  is  lost  by  paying  the  second  debt 
before  it  is  due  ;  but  the  gain  arising  from  the  keeping  of  a  sum 
of  money  after  it  is  due,  is  evidently  equal  to  the  interest  of  the 
debt  for  that  time  ;  and  tbe  loss  which  is  sustained  by  the  paying  ' 
of  a  sum  of  money  before  it  is  due,  is  evidently  the  discount  of 
the  debt  for  that  time  ;  therefore  it  is  obvious  that  the  debtor  must 
retain  the  sum  immediately  due,  or  the  first  payment,  till  its  inter- 
est shall  be  equal  to  the  discount  of  the  second  sum  for  the  time 
it  is  paid  before  due;  because  in  that  case  the  gain  and  loss  will 
be  equal,  and  consequently  neither  party  can  be  a  loser. 

REVIEW. 

For  what  purpose  is  equation  used  ?  What  is  the  first  rule  ? 
The  second  1  What  can  you  say  in  relation  to  the  correctness 
of  the  principle  of  calculating  equations  of  time  of  payment  ? 


BARTER. 

This  is  a  rule  by  which  merchants  and  others  exchange  one 
commodity  for  another,  and  by  which  they  know  how  to  make 
the  exchange,  or  proportion  the  quantities  without  loss  to  either 
party ;  the  rules  in  Barter,  Profit  and  Loss,  and  Partnership, 
are  only  applications  of  the  rules  of  proportion  which  have  been 
explained  and  are  easily  understood. 

RULE    I. 

Find  the  value  of  what  you  propose  to  exchange  at  the  price 
at  which  you  wish  to  exchange  it,  by  any  rule  most  convenient. 
Then,  as  the  price  of  one  of  the  articles  which  you  receive,  is  to 
the  whole  quantity,  so  is  the  whole  value  of  what  you  give  in 
exchange,  to  the  answer  required. 

QUESTIONS. 

1.  A.  has  200  pounds  of  tea  at  D1.25  per  pound,  which  he 
will  let  B.  have  in  exchange  for  sugar  at  lie.  per  pound  ;  how 
much  su^ar  must  A.  receive  for  his  tea  ? 


I 


D1.25  per  lb.  tea. 
200  lb.  tea. 


c.ll)250.00(2272lb.  II0Z.+  A71S. 
Or,  as  lie.  :  2001b.  ::   Dl.25  :  22721b.  lloz. 

2.  A.  and  B.  barter,  A.  sold  B.  4.5  yards  of  broadcloth  at 
D5.25  per  yard,  150  lb.  of  pork  at  8.5  per  lb.,  and  one  barrel  of 
mackerel  at  D10.25  ;  B.  let  A.  have  2.5  cords  of  M^ood  at  D3.75 
per  cord,  10  bushels  of  wheat  at  Dl.25  per  bushel,  and  7  bush 
els  of  rye  at  80c.  per  bushel ;  how  does  the  account  stand  ? 

Ans.  B.  must  pay  to  A.  D19.15. 

3.  D.  sold  a  horse  for  D75,  one  half  was  paid  in  cash  down 
and  the  remainder  in  oats  at  37.5c.  per  bushel ;  how  many 
bushels  of  oats  did  he  receive  ?  Ans.  100. 

4.  How  much  corn  at  87.5c.  per  bushel,  is  equal  to  464  bush- 
els of  wheat  at  Dl.25  per  bushel  ? 

Ans.  662  bushels,  3  pecks,  3  quarts. 

5.  How  much  tallow  at  8.5c.  will  you  receive  in  barter,  for 
12cwt.  2qrs.  81b.  of  sugar  at  15c.  per  lb.  ?     Ans.  24841b,  lloz. 

6.  C.  has  2  pieces  of  broadcloth  ;  1  piece  contains  30  yards 
at  D4.50  per  yard  ;  the  other  25  yards  at  D3.75  per  yard,  which 
he  will  barter  with  P.  for  20001b.  of  pork  at  10.5c.  per  lb.  and 
the  balance  in  flax  at  12^c.  per  lb. ;  how  much  flax  will  C.  re- 
ceive ?  Ans.  1501b. 

7.  W.  has  3  hogsheads  of  wine  at  Dl.12  per  gallon,  for  tho 
value  of  one  half  he  will  take  wheat,  at  Dl.lO  per  bushel,  and 
for  the  remainder  he  will  take  250  yards  of  domestic  cloth  ;  how 
much  wheat  will  he  receive,  and  how  much  will  the  cloth  cost 
him  per  yard  ? 

Ans.  96  bushels,  6  quarts,  1  pint,  wheat ;  and  cost  of  cloth, 
c.42.34-  per  yard. 

8.  How  much  butter  at  14c.  per  pound  must  be  given  for  85 
pounds  of  ham  at  16c.  per  pound,  and  8  pounds  of  tea  at  Dl.25 
per  pound  ?  Ans.  1681b.  9oz. 

9.  Two  farmers  bartered  ;  A.  had  120  bushels  of  wheat  at 
Dl.50  per  bushel,  for  which  B.  gave  him  100  bushels  of  barley 
worth  65c.  per  bushel,  and  the  balance  in  oats  at  40c.  per  bush- 
el ;  what  quantity  of  oats  did  A.  receive  ?         Ans.  287.5  bush. 

10.  A  merchant  sold  14.5  yards  of  broadcloth  at  D4.25  per 
yard,  and  received  in  payment  95lb.  of  wool  at  31.25c.  per  lb., 
15doz.  of  eggs  at  16c.  per  dozen,  4.25  bushels  of  wheat  at  Dl  .12 
per  bushel,  a  quarter  of  beef  weighing  1641b.  at  6-^c.  per  lb.,  471b. 
of  tallow  at  12^c,  per  lb.,  8  bushels  of  rye  at  93^0.  per  bushel ; 

15 


h 


170  BARTER. 

and  78lc.  in  cash  ;  how  much  more  must  he  (A  )  receive  loi 
his  cloth  ?  Ans.  0. 

11.  A.  and  B.  bartered  ;  A.  had  8.25  cwt.  of  sugar  at  12  cts. 
per  lb.,  for  which  B.  gave  him  18  cwt.  of  flour  ;  how  much  was 
the  flour  per  lb.  ?  Ans.  5.5c. 

12.  B.  has  5  pieces  of  muslin,  each  piece  containing  95  yards 
at  23c.  per  yard,  for  which  C.  is  to  give  him  32  sheep  at  D2.50 
each,  and  the  remainder  in  rye  flour  at  Dl.50  per  cwt.  ;  how 
many  cwt.  of  flour  must  C.  receive  ?  Ans.  19cwt.  2qr. 

13.  A.  purchased  a  flock  of  sheep  of  B.  consisting  of  75  in 
number,  at  D1.75  each  ;  he  paid  B.  D87.50  in  cash,  1.5  tons  of 
hay  at  D7.50  per  ton,  and  9  bushels  of  corn  at  62ic.  per  bush- 
el ;  required  the  balance  due.  A7is.  1)26.87.5. 

Having  the  ready  money  and  bartering  price  of  one  article 
given  and  the  ready  money  price  and  quantity  of  another  article 
given,  to  find  the  bartering  price  and  the  quantity  of  the  other. 


As  the  ready  money  price  of  the  one  :  is  to  the  ready  money 
price  of  the  other  : :  so  is  the  bartering  price  of  the  first  :  to 
the  bartering  price  of  the  second. 

14.  A.  and  B.  barter  ;  A.  has  145  gallons  of  wine  at  Dl.20 
per  gallon  ready  money,  but  in  barter  will  have  Dl.35  per  gal- 
lon ;  B.  has  Hnen  at  58c.  per  yard,  ready  money;  how  must  B. 
sell  his  linen  per  yard,  in  proportion  to  A.'s  bartering  price,  and 
ho.v  many  yards  are  equal  in  value  to  A.'s  win©  ? 

Ans.  65.25c.  linen  per  yard,  and  300  yards. 

15.  A.  has  wheat  worth  Dl.13  per  bushel  ready  money,  but 
in  barter  he  will  have  D1.33  per  bushel ;  B.  has  butter  worth  20c. 
per  lb.  ready  money;  how  must  B.  rate  the  butter  to  be  equal 
to  the  wheat  ?  A7is.  23.5c. 

16.  P.  has  indigo  worth  Dl.OO  per  pound  ready  money,  but  in 
barter  he  will  have  Dl.13  per  pound  ;  C.  has  sugar  worth  10c. 
per  pound  ready  money  ;  how  must  C.  rate  his  sugar  that  it  may 
be  equal  Avith  the  indigo?  Ans.  11.3c. 

17.  A.  has  coffee  worth  20c.  per  lb.  in  ready  money,  but  in 
barter  he  will  have  25c.  per  lb. ;  13  has  broadcloth  worth  D2  per 
yard  ready  money  ;  at  what  price  ought  the  broadcloth  to  be  ra- 
ted in  barter  1  Ans.  D2.50. 

18.  A.  has  320  doz.  of  candles  at  Dl.20  per  dozen,  for  which 
B.  agrees  to  pay  him  D160  in  cash,  and  the  rest  in  cotton  at  20c. 
per  lb  I  how  much  cotton  must  B.  give  A.  1 

Ans.  n20lb. 


BARTER.  ^  l1 

19.  A  person  purchased  120  tons  of  iron  at  D  10.25  per  toil 
and  paid  as  follows,  namely  :  In  cash  D500  ;  27  bushels  of  sali 
at  65c.  per  bushel ;  1501b.  of  leather  at  25c.  per  lb. ;  8  bushelsX 
of  clover-seed  at  D4.75  per  bushel  ;  15  bushels  of  flax-seed  at' 
D  1.10  per  bushel ;    75  gallons  of  currant  wine  at  Dl.25  per 
gallon  ;  250  bushels  of  oats  at  37.5  per  bushel  ;    and  he  is  to 
pay  the  balance  in  honey  at  75c.  per  gallon  ;  required  the  bal- 
ance due,  and  the  quantity  of  honey  to  be  paid. 

Arts.  Due  D432.95  ;  honey  577  gallons,  1  quart. 

20.  H.  has  75  sheep  at  Dl.75  each,  for  which  VV.  is  to  give 
him  D109  in  cash,  and  the  rest  in  corn  at  62.5c.  per  bushel; 
how  much  corn  must  W.  give  H.  ? 

Ans.  35  bushels,  2  pecks,  3  quarts. 

21.  P.  and  Q.  barter;  P.  has  Irish  linen  at  60c.  per  yard,  but 
in  barter  he  will  have  64c.  per  yard  ;  Q.  delivers  him  broad- 
cloth at  D6  per  yard,  worth  only  1)5.50  per  yard  ;  which  has  the 
advantage  in  the  bargain,  and  how  much  linen  does  P.  give  Q. 
for  148  yards  of  broadcloth  ? 

As  60  :  64  ::  5.50  :  5.86|- ;  therefore,  Q.  by  selling  at  D6 
has  the  advantage  ;  then  6  :  148  yards  ::  64  :  1578|  yds.  Ans. 

22.  A.  has  linen  cloth  at  30c.  per  yard  ready  money,  in  bar- 
ter 36c.,  B.  has  3610  yards  of  riband  at  22c.  per  yard  ready 
money,  and  would  have  of  A.  D200  in  ready  money,  and  the 
rest  in  linen  cloth  ;  what  rate  does  the  riband  bear  in  barter  per 
yard,  and  how  much  linen  must  A.  give  B.  ?  The  rate  of  rib- 
and is  26.4c.  per  yard,  and  B.  must  receive  1980|-  yards  of  lin- 
en and  D200  in  cash. 

23.  A.  has  150  yards  of  linen,  at  25c.  per  yard,  which  he 
wishes  to  exchange  with  B.  for  muslin  at  50c.  per  yard  ;  how 
much  muslin  must  A.  receive  1  Ans.  75  yards. 

24.  A.  had  200  barrels  of  flour  at  D  10.50  per  barrel,  for  which 
B.  gave  him  D1090  in  money,  and  the  rest  in  molasses  at  20c. 
per  gallon  ;  how  many  hogsheads  of  molasses  did  he  receive  ? 

Ans.  80  hogsheads,  }0  gallons. 

25.  A  merchant,  in  bartering  with  a  farmer  for  wood  at  Do  per 
cord,  rated  his  molasses  at  D25  per  hogshead,  which  was  worth 
no  more  than  D20 ;  what  price  ought  the  farmer  to  have  a  cord 
for  his  wood  to  be  equal  to  the  merchant's  bartering  price  ? 

Ans.  D6.25. 

26.  A  farmer  sold  a  grocer  20  bushels  of  rye  at  75c.  per  bush- 
1 ;  2001b.  of  cheese  at  10c.  per  pound ;  in  exchange  for  which 

he  received  20  gallons  of  molasses,  at  22c.  per  gallon,  and  the 
balance  in  money ;  how  much  money  did  he  receive  ? 

Ans.  D30.60 


172  PROFIT  AND  LOSS. 


REVIEW. 


What  is  barter  ?  What  will  you  first  do  when  you  wish  to 
barter  one  commodity  for  another?  What  is  the  first  rule? 
After  having  found  the  value  of  the  article  you  wish  to  exchange, 
how  will  you  find  the  answer  ?  Repeat  the  second  rule,  and 
explain  its  operation. 


PROFIT  AND  LOSS. 

This  rule  is  used  to  find  the  profit  or  loss  sustained  in  the 
purchase  or  sale  of  property  of  every  description,  and  to  ad- 
vance or  reduce  the  value  of  any  article  so  as  to  gain  or  lose  so 
much  per  cent. 

When  property  or  an  article  is  purchased  at  a  certain  price,  and 
is  to  be  sold  at  any  other  price,  either  more  or  less,  to  ascertain  the 
profit  or  loss  on  the  whole  work  by  the  following  rule. 


1.  Find  the  whole  amount  of  the  cost.  2.  Find  the  amount 
it  sold  for.  3.  Then  if  the  sum  it  sold  for  be  more  than  the 
cost,  subtract  the  sum  it  cost  from  the  sum  it  sold  for,  and  the 
remainder  will  be  the  profit.  4.  But  if  it  sold  for  less  than  it 
cost,  subtract  the  sum  it  sold  for  from  the  sum  it  cost,  and  the 
remainder  will  be  the  loss. 

QUESTIONS. 

1.  A.  purchased  501b.  of  cheese  at  12.5c.  per  pound,  and  sold 
it  for  15c,  per  pound  ;  what  is  the  profit  on  the  whole  ? 

Thus  :  price  per  lb.  12.5  cents.  15  cents,  price  it  sold  for. 

50  pounds.  50  pounds. 

^  D6.25.0  cost.         D7.50  sold  for. 

*-6.25 

Di.25  profit. 


•  PROFIT  AND  LOSS.  17S 

Or  thus,  15c. 

—  12.5c. 


2.5  gain  per  pound. 
50  pounds. 


Dl.25  profit. 

2.  Purchased  12961b.  of  sugar  at  9c.  per  pound,  and  sold  it 
at  11.5c.  per  pound  ;  what  is  the  profit  on  the  whole  ? 

Ans.  D32.40. 

3.  When  250  bushels  of  wheat  are  purchased  at  Dl.25  per 
bushel,  and  sold  for  D  1.3 1.25  per  bushel;  what  is  the  profit  on 
the  whole  1 

Ans.  D15.62.5. 

4.  If  you  purchase  180  bushels  of  rye  at  D0.93|  per  bushel, 
and  sell  it  for  Dl.02  ;  what  is  the  profit  on  the  whole  ? 

Ans.  D15.75. 

5.  If  1500  pounds  of  liam  cost  12.5  cents  per  pound,  and  is 
sold  for  18.75  ceiits  per  pound,  what  is  the  profit  on  the  whole  ?' 

■    Ans.  D93.75. 

6.  If  a  merchant  should  purchase  1150  bushels  of  corn,  at 
87.5c.  per  bushel,  and  sell  it  at  85c.  per  bushel,  what  would  his 
loss  be  on  the  whole  ?  Ans.  D28.75. 

7.  If  you  purchase  25  barrels  of  flour  at  D9  per  barrel,  and 
sell  it  for  5c.  per  pound,  will  you  gain  or  lose,  and  how  much  ? 

Ans.  D20,  giin. 

8.  If  a  hogshead  of  sugar  weighing  9c wt.  2qr.  cost  D120,  and 
be  sold  for  10c.  per  pound,  what  is  gained  or  lost  on  the  whole  ? 

Ans.  D13.60,  lost. 

9.  A  merchant  purchased  30  ydrds  of  broadcloth  at  1)4.75 
per  yard,  and  sold  it  for  D5.06.75  ;  required  the  profit  on  the 
whole.  ,  Ans.  D9.52.5. 

10.  A.  paid  D 130 for  a  pipe  of  wine,  which  he  sold  for  Dl.02 
per  gallon ;  did  he  gain  or  lose,  and  how  much  on  the  whole  ? 

Ans.  ri.48,  lost. 

When  property  is  purchased  or  soldy  to  know  what  is  gavned  or 
lost  per  cent. 


The  price  it  cost  is  the  first  term  ;  the  difference  of  what  it 
cost  and  what  it  sold  for  is  the  second  term;  and  100  is  the 
third  term ;  then  work  by  simple  proportion. 

15* 


174  TROFIT  AND   LOSS-. 

11.  Bought  120  bushels  of  oats  at  50c.  per  bushel,  and  sold 
them  for  55c.  per  bushel ;  what  was  the  gain  on  the  whole,  and 
gain  per  cent.  ? 

Thus  :   120  bushels.  then  120  bushels. 

50c.  cost  per  bustiel.  55c. 


D60.00  cost.  D66.00  sold  for. 

—  60.00 


D6.00  gained  on  the  whole 

Or:      120  bushels.  then  55 

5c.  gain  on  1  bushel.      —50 

D6.00  A71S.  50  :  5c.  ::  100  li 

5  :'• 

60)500  ' 

10  per  cent. 

12.  Bought  a  yard  of  linen  cloth  for  37.5c.  and  sold  it  for  32c. , 
what  is  the  loss  per  cent.  ?  Ans.  14. 6c.  per  cent. 

13.  If  I  pay  25  cents  per  pound  for  butter,  and  purchase  550 
pounds,  then  sell  the  same  for  31  cents  per  pound,  what  do  I 
gain  on  the  whole,  and  how  much  per  cent.  ? 

Ans.  D33,  and  24  per  cent. 

14.  A.  paid  D225  for  200  pounds  of  tea,  and  sold  it  for  Dl.05 
per  pound  ;  did  he  gain  or  lose  ?  and  how  much,  and  what  per 
cent.  1  Ans.  Lost  D15,  which  is  6|-  per  cent. 

15.  Bought  150  pounds  of  pork  for  Dll.oO,  and  1  wish  to 
gain  D2.50  profit ;  how  much  must  I  sell  it  i'or  per  pound  1 

Ans.  9c.  3m. 

16.  Purchased  200  pounds  of  tallow  for  D18,  and  sold  it  for 
D25;  how  much  did  I  get  per  pound,  and  how  much  profit  per 
cent.  ?  e^/z.s'.  12.5c,  per  pound,  and  89  per  cent. 

TTie  prime  cost  and  the  profit  or  loss  per  cent,  given,  to  find 
what  it  sold  for. 


DlOO  is  the  first  term  ;  the  prime  cost  is  the  second  term ; 
and  DlOO,  wnth  the  gain  added,  or  loss  subtracted,  is  the  third 
term  ;  and  the  quotient  is  the  answer. 

17.  If  a  bushel  of  corn  cost  50c.,  what  must  it  be  sold  for  io 
^in  25  per  cent.  ?  Thus:   100  :  50  ::   125  :  62.5c.  Ans, 


PROFIT  AND  LOSS.  175 

18.  If  I  purchase  sheeting  at  8c.  per  yard,  at  what  price 
must  it  be  sold  to  gain  25  per  cent.  ?  Ans.  10c. 

19.  At  25c.  profit  on  a  dollar,  how  much  per  cent.  ? 

Ans.  25  per  cent. 

20.  If  I  purchase  wine  at  Dl.05  per  gallon,  how  must  I  sell 
it  to  gain  30  per  cent.  ?  Ans.  Dl.36.5. 

When  the  profit  or  loss  per  cent,  is  given,  with  the  selling  price  oj 
the  article,  to  find  the  prime  cost. 

RULE  IV. 

DlOO  with  the  gain  per  cent,  added,  or  loss  per  cent,  subtract- 
ed, is  *he  first  term  ;  DlOO  is  the  second  term  ;  the  prime  cost 
is  the  thiil  term. 

21.  If  by  selling  wheat  at  D5  there  is  25  per  cent,  gained, 
what  is  the  first  cost  ? 

Thus:   100+25=125  :   100  ::  5  :  D4..  Ans. 

22.  If  in  purchasing  cloth  at  D30  there  is  10  per  cent,  gain- 
ed, what  was  the  prime  cost?  Ans.  D27.27.2. 

23.  If  I  sell  a  bushel  of  oats  for  45c.  and  lose  10  per  cent., 
required  the  prime  cost.  Ans.  50c. 

24.  If  I  sell  rye  at  87.5c.  and  lose  12.5  per  cent.,  how  much 
did  it  cost?  Ans.Dl.QO. 

The  selling  price  and  the  gain  or  loss  per  cent,  given,  to  find  what 
would  be  the  profit  or  loss  per  cent,  if  sold  at  any  other  price. 

RULE    V. 

The  first  price  is  the  first  term  ;  the  second  price  is  the  sec- 
ond term  ;  DlOO  with  the  gain  per  cent,  added  or  loss  per  cent, 
subtracted,  is  the  third  term  ;  the  quotient  is  the  answer. 

25.  A  person  sold  a  yard  of  cloth  for  D2.23,  and  gained  10 
per  cent. ;  if  he  had  sold  it  for  D2.75,  what  would  have  been 
the  gain  per  cent.  ? 

Thus:  2.23  :  2.75  ::  110  :  D135.65— Dl00=35|  per  ct.  Ans. 

26.  Bought  a  hogshead  of  molasses  containing  119  gallons  at 
52c.  per  gallon,  paid  for  transportation  Dl.21,  and  by  accident 
9  gallons  leaked  out ;  at  what  rate  must  I  sell  the  remainder  per 
gallon  to  gain  Dl3  on  the  first  cost?  Ans.  69.17c. 

27.  If  I  purchase  12cwt.  2qrs.  of  rice  at  D3.45  per  cwt.  and 
sell  it  at  4c.  per  pound,  what  is  my  profit?  Ans.  D12.87.5. 

28.  Bought  Icwt.  of  cotton  for  D34.86  and  sell  it  for  41.5 
per  pound  ;  what  do  I  gain  or  lose,  and  what  per  cent.  ? 

Ans.  Gain  D11.62,  and  33.3  per  cent. 


176  PROFIT  AND  LOSS. 

29.  If  by  selling  cloth  at  D3.25  per  yard,  I  lose  at  the  rate  of 
20  per  cent.,  what  is  the  prime  cost  of  said  cloth  per  yard  ? 

Ans.  D4.06.25 

30.  Bought  a  chest  of  tea  weighing  4751b.  for  D420  and  sole* 
the  same  for  D350 ;  how  much  did  I  lose  on  the  pound  ? 

Ans.  14.7c. 

31.  Bought  cloth  at  Dl.25  per  yard,  which  proving  bad,  1 
wished  to  sell  it  at  a  loss  of  18  per  cent.  ;  hpw  much  must  I  ask 
per  yard  ? 

32.  Bought  a  cow  for  D30  cash,  and  sold  her  for  D35  at  a 
credit  of  8  months  ;  reckoning  interest  (on  D35)  at  6  per  cent. 
how  much  did  I  gain  ? 

33.  A  merchant  buys  158  yards  of  calico  for  which  he  pays 
20  cents  per  yard  ;  one  half  is  so  damaged  that  he  is  obliged  to 
sell  it  at  a  loss  of  6  per  cent.,  the  remainder  he  sells  at  an  advance 
of  19  per  cent.  ;  how  much  did  he  gain  ?  Ans.  D2.054- 

34.  A  tailor  bought  60  yards  of  cloth  at  D4.50  per  yard,  and 
38  yards  at  D2.50  per  yard,  and  sold  the  whole  quantity  at 
D4.25  per  yard  ;  did  he  gain  or  lose,  and  what  per  cent.  1 

Ans,  Gained  D51.50,  and  Dl4.ll  gained  per  ct.  (on  DlOO). 

35.  A.  purchased  cloth  at  D2.50  per  yard,  but  being  damaged 
he  is  willing  to  lose  17-|  per  cent,  on  it ;  how  must  he  sell  it 
per  yard?  \  Ans.  1)2.06.25. 

36.  If  I  buy  a  bale  of  cotton  weighing  8cwt.  201b.  at  a  cost  of 
D45.55,  how  must  it  be  sold  per  cwt.  to  lose  8  per  cent,  by  it  ? 


What  do  you  understand  by  profit  and  loss  ?  In  what  cases 
can  you  apply  the  several  rules  ?  When  an  article  is  purchased 
at  a  certain  price,  and  is  sold  at  any  other  price,  how  will  you 
ascertain  the  profit  or  loss  ?  Repeat  rule  1.  When  property  is 
sold,  to  know  what  is  gained  or  lost  per  cent.,  how  will  you  pro- 
ceed ?  What  is  rule  2  ?  What  will  you  do  when  the  prime  cost 
and  the  profit  or  loss  per  cent,  are  given,  to  find  what  it  sold  for  ? 
Repeat  rule  3,  and  give  an  example  in  this  rule.  When  the 
profit  or  loss  per  cent,  is  given  with  the  selling  price  of  the  arti- 
cle, to  find  the  prime  cost,  what  is  to  be  done  ?  What  is  rule  4  ? 
When  the  selling  price  and  the  gain  or  loss  per  cent,  are  given, 
to  find  what  would  be  the  profit  or  loss  per  cent,  if  sold  at  any 
other  price  what  is  to  be  done  ?  Repeat  rule  5.  Now  solve 
the  following  question  : — 

37.  I  sold  a  watch  for  D50,  and  by  so  doing,  lost  D17  pei 
cent.,  whereas  in  trading  I  ought  to  have  cleared  D20  per  cent. ; 
how  much  was  it  sold  under  its  real  v-alue  ?         Ans.  D22.28.8. 


-  PARTNERSHIP.  177 

PARTNERSHIP. 

Partnership,  or  as  it  is  frequently  termed,  fellowship,  is  a 
rule  used  by  merchants,  tradesmen,  and  others  who  deal  or 
transact  business  in  company,  or  partnership,  to  ascertain  the:r 
equitable  share  of  profit  or  loss  when  their  proportion  of  stock 
is  unequal,  or  when  the  stock  has  been  invested  for  a  longer  or 
shorter  time  ;  and  also  by  this  rule  the  estate  of  an  insolvent 
may  be  justly  divided  between  his  creditors,  and  the  equitable 
distribution  of  property  of  every  description,  the  assessment  of 
taxes,  &c.,  (fee.  The  money  employed  is  the  capital  stock. 
The  gain  or  loss  to  be  shared  is  called  the  dividend.  The  op- 
eration of  the  rules  is  proportional. 

Wheji  time  is  not  considered 
rule   I. 

The  amount  of  stock,  or  whole  debt,  is  the  first  term  ;  either 
the  partners'  shares  or  sum  in  trade  is  the  second  term  (for 
his  share)  ;  the  whole  gain  or  loss  is  the  third  term  ;  and  the 
quotient  is  the  amount  of  that  partner's  share  of  the  profit  or 
loss  ;  and  so  continue  the  statement  for  each  partner,  respec- 
tively. Proof:  add  the  shares  of  the  profit  or  loss  of  each  of 
the  partners  together,  and  if  the  amount  agrees  with  the  sum 
mentioned  in  the  question,  it  will  be  correct. 


1 .  Divide  the  whole  gain  or  loss  by  the  whole  stock,  and  the 
quotient  will  be  the  gain  or  loss  per  dollar. 

2.  Multiply  the  gain  or  loss  per  dollar  by  each  particular 
stock,  and  the  product  is  the  proportional  gain  or  loss  required. 

QUESTIONS. 

1.  A.  and  B.  formed  a  connexion  in  business  ;  A.  advanced 
D750,  B.  D250  ;  after  trading  together,  they  find  their  profits 
amount  to  D400  ;  required  a  just  division  of  their  gain. 

A.  B. 

A.  750  >  1000  :  750  ::  400  G.  (  1000  :  250  ::  400   )  300  A. 

B.  250  J        400      I  400      5  100  B. 

1000  S  1000)300000(300  A.  \  1000)100000(100  B.  \  400  Pf. 

2.  A.  and  B.  purchased  goods  to  the  amount  of  DlOOO;  A. 
paid  D600,  and  B.  D400  ;  they  gained  DlOO  ;  required  the 
share  of  each. 


178  PARTNERSHIP. 

Rule  2d.  600  A.  )  A.  600  x  10  =  60  A.        >  share? 

400  B.  S  B.  400 X  10  =  40  B.        5      Ans 

gain  D]00)1000(10c.  per  D.  \  100  proof.  \ 

3.  A.,  B.,  C,  and  D.,  formed  a  partnership  :  A.  put  in  D3200, 

B.  D2100.  C.  D2150,  and  D.  D550  ;  on  examination  they  find 
their  gain  to  be  D2500  ;  what  is  the  share  of  each  ? 

Ans,  A.  DIOOO;  B.  D656.25  ;  C.  D671.87.5  ;  D.  D171.87.5. 

4.  A.,  B.,  and  C,  freighted  a  ship  with  108  tons  of  wine,  of 
which  A.  had  48  T.,  B,  36  T.,  and  C  24  T. ;  but  in  stress  of 
weather  they  were  obliged  to  throw  45  T.  overboard  ;  how 
much  must  each  sustain  of  the  loss  ?  An^.  A.  20,  B.  15,  C.  10  T. 

5.  Three  farmers,  A.,  B.,  and  C,  occupy  a  farm  of  350  acre.s, 
of  which  A.  has  100  acres,  B.  110  acres,  C.  140  acres,  for  which 
thev  pay  D750  ;  what  must  each  man  pay  in  proportion  to  the 
land  he  holds  ?     Ans.  A.  D214.28.5  ;  B.  D235.71.4  ;  C.  D300. 

6.  Four  men  traded  with  a  stock  of  D800,  by  which  they 
gained  D307 ;  A.'s  stock  was  DUO,  B.'s  D260,  C.'s  D300  ; 
required  D.'s  stock,  and  what  each  man  gained. 

Ans.  D's  stock  DlOO  ;   A.  gained  D53.72.5  ;  B.  D99.77.5  ; 

C.  D115.12.5;  D.  D38.37.5. 

7.  A.,  B.,  and  C,  traded  in  partnership  ;  A.  put  in  D140, 
B.  D250,  and  C.  put  in  120  yards  of  cloth  at  cost  price  ;  they 
gained  D230,  of  which  C.  took  DlOO  for  his  share  of  the  gain  ; 
how  did  C.  value  his  cloth  per  yard  in  common  stock,  and  what 
was  A.  and  B.'s  part  of  the  gain  ? 

Ans.  C's  cloth  D2.50  per  yard;  A.  D46.66.6+  ;  B. 
D83.33.3+  ;  C.  share,  D.  100. 

8.  Three  merchants  trading  together  lost  goods  to  the  amount 
of  D1920  ;  suppose  A.'s  stock  was  D2880,  B.'s  D11520,  C.'s 
D4800  ;  what  share  of  the  loss  must  each  sustain  ? 

Ans.  A.  D288 ;  B.  D1152  ;  C.  D480. 

9.  A.,  B.,  C,  and  D.,  gain  D200  in  trade,  of  which  as  often 
as  A.  has  D6,  B.  must  have  DlO,  C.  Dl4,  and  D.  D20 ;  what 
is  the  share  of  each  ?  Ans.  A.  D24  ;  B.  D40  ;  C.  D56  ;  D.  D80. 

10.  An  insolvent  estate  of  D633.60  is  indebted  to  A.  D3 12.75, 
o  B.  D297.00,  to  C.  D50.25,  to  D.  D0.25,  to  E.  D200,  to  F. 

D  142.50,  and  to  G.  D21.25  ;  what  proportion  will  each  credi- 
tor receive  ? 

Ans.  A.  D193.51.41;  B.  Dl83.76.87;  C.  D31.09.23  ;  D 
DO. 15.41;  E.  D123.75 ;  F.  D88.17.18;  G.  D13.14.87  = 
D633.59.97+  proof. 


». 


PARTNERSHIP.  179 

11.  A.,  B.,  and  C,  put  their  money  into  joint  stock  ;  A.  put 
in  D40,  B.  and  C.  together  D170  ;  they  gained  D126,  of  which 
B.  took  D42  ;  what  did  A.  and  C.  gain,  and  B.  and  C.  put  in 
respectively  ? 

Ans.  A.'s  gain  D24,  B.'s  stock  D70,  C.'s  stock  DlOO,  and 
C.'s  gain  D60. 

12.  A.  and  B.  companied ;  A.  put  in  D45,  and  took  f  of  the 
gain  ;  what  did  B.  put  in  1  Ans.  D30. 

13.  A.  and  B.  venture  equal  stocks  in  trade,  and  clear  D164; 
hy  agreement  A.  was  to  have  5  per  cent,  of  the  profits,  because 
he  managed  the  concerns  ;  B.  was  to  have  but  2  per  cent. ;  what 
was  each  one's  gain  ?  how  much  did  A  receive  for  his  trouble  ? 

Ans.  A.'s  gain  was  Dll7.142-f,  and  B.'s  P46.857J,  and  A. 
received  D70.285^  for  his  trouble. 

TAXING. 

A  tax  is  a  sum  required  to  be  paid  to  the  government  for  its 
pport,  or  for  other  purposes,  and  is  generally  collected  from 
each  individual  in  proportion  to  his  property ;  this,  however,  is 
different  in  some  states,  where  every  white  male  citizen  over  the 
age  of  21  years  is  required  to  pay  a  certain  tax,  called  a  poll- 
tax  ;  in  this  case  each  person  is  called  a  poll :  hence  the  expres- 
sions, *'  polling  votes,"  "  attending  the  polls,"  &;c.  In  the  assess- 
ment of  taxes,  we  must  first  make  an  inventory  of  all  the  prop- 
erty, both  real  and  personal,  of  the  whole  town  or  district  to  be 
taxed,  and  also  of  each  individual  who  is  to  be  taxed  ;  and  as 
the  number  of  polls  are  rated  at  so  much  each,  the  tax  on  all  the 
polls  must  be  taken  out  from  the  whole  tax,  and  the  remainder 
is  to  be  assessed  on  the  property.  Then  to  find  how  much  any 
individual  must  be  taxed  for  his  property,  we  need  only  find 
how  much  the  remainder  of  the  whole  tax  is  on  Dl,  and  multi- 
ply his  inventory  by  it. 

14.  If  a  town  raise  a  tax  of  D1920,  and  all  the  property  in 
town  be  valued  at  D64000,  what  will  that  be  on  Dl,  and  what 
will  be  A.'s  tax  whose  property  is  valued  at  D1200  ? 

Ans.  DO. 03c.  on  a  dollar;  or  r36.00  :  for  Dl920-^64000= 
3c.xDl200=36.00  A's  tax. 

15.  A  certain  town,  valued  a  D64530,  raises  a  tax  of 
D2259.90;  there  are  540  polls  which  are  taxed  D60  each; 
what  is  the  tax  on  a  dollar,  and  what  will  be  A.'s  tax,  whoso 
real  estate  is  valued  at  D1340,  his  personal  property  at  D874, 
and  who  pays  for  2  polls  ? 

Thus,  D540  X  .60=:D324  amount  of  the  poll-taxes  ;  D2259.90 
--D324=D1 935.90,  to  be  assessed  on  property  ;  then  64530  : 
1935  ::  1  :  .03,  or  1935.90^64530  =  .03,  or  3c.,  tax  Dl. 


^ 


180 


PARTNERSHIP. 


Tax  on  Dl  is  .03 

2        .06 


3 
4 
5 
6 

7 
8 
9 


.09 
.12 

.15 
.18 
.21 
.24 
.27 


TABLE. 
Tax  on  DIO  is  .30  Tax  on  D 100  is  D3 


20 
30 
40 
50 
60 
70 
80 
90 


.60 
.90 
1.20 
1.50 
1.80 
2.10 
2.40 
2.70 


200 

6 

300 

9 

400 

12 

500 

15 

600 

18 

700 

21 

800 

24 

900 

27 

1000 

30 

Now,  to  find  A.'s  tax,  his  real  estate  being  D1340  ;  I  find  by 
the  table,  that  the  tax  on  DlOOO  is         D30 
The  tax  on        300  is  9 

The  tax  on         40  is  1 .20 


Tax  on  his  real  estate 
In  like  manner,  I  find  the  tax  on 
his  personal  property  to  be 

2  polls  at  .60  each  are 


D40.20 

D26.22 

1.20 


Amount,     D67.62  Ans, 


When  the  stocks  are  considered  with  regard  to  time. 


RULE    III. 


1.  Multiply  each  man's  stock  by  its  time,  and  add  the  prod* 
ucts  together. 

2.  Then,  as  the  sum  of  the  whole  stock,  multiplied  by  the 
time,  is  to  the  product  of  each  individual  share,  multiplied  by 
its  time,  so  is  the  whole  gain  or  loss  of  each  individual. 

16.  Three  merchants  traded  together  for  12  months  ;  A.  put 
in  D120  for  4  months,  B.  put  in  D250  for  9  months,  and  C.  put 
in  D300  for  12  months,  they  gained  D175  ;  what  will  each  man 
receive  for  the  gain  ? 

A.  120  X   4mo.=  480  )  A.  6330  :    480  ::  175g.  D13.27  Ans 
B.250X   9    "   =2250  \  B.  6330  :  2250  ::  175"      62.20.4 
C.  300x12    "   =3600lC.  6330  :  3600  ::  175"      99.52.6 

D175,00.0  Pf, 


PARTNERSHIP.  ~  181 


RULE    IV. 


1st.  Multiply  each  man's  stock  by  the  time  it  was  in  trade, 
for  its  product. 

2d.  Divide  the  gain  or  loss  by  the  sum  of  the  products  for 
their  gain  or  loss  per  dollar. 

3d.  Multiply  the  gain  or  loss  per  dollar,  by  each  product,  for 
each  man's  proportional  gain  or  loss,  and  the  product  is  the 
answer. 

17.  A.  and  B.  traded  12  months  ;  A.  put  in  D300  for  12 
months,  B.  put  in  D600  for  6  months,  their  gain  is  D288  ;  re- 
quired the  share  of  each. 

A.  DSOOx  12=3600  >      72)288(.04c.  per  D, 

B.  600  X    6  =  3600 


7200 


A.  3600  B.  3600  A.  144.00 

4  4  B.  144.00 


D144.00  D144.00  D288.00  Proof. 

18.  A.  and  B.  formed  a  partnership  for  18  months  ;  A.  put 
in  D2750  for  15  months,  and  B.  D3500  for  10  months  ;  at  the 
expiration  of  that  time  he  took  out  D2000  ;  at  the  close  of  their 
business  they  find  their  gain  to  be  D1500  ;  what  is  each  man's 
share  of  the  gain  ? 

Ans,  A.'s  D701.13.3+;  B.'s  D798.86.7+. 

19.  A.  put  in  stock  to  the  amount  of  D1800,  B.  advanced  4 
mo.  after  ;  what  sum  must  he  put  in  to  receive  equal  profit  with 
A.  at  the  end  of  the  year  ?  A?is.  D2700. 

20.  A.  and  B.  formed  a  partnership;  A.  put  in,  the  1st  of 
January,  DlOOO,  but  B.  could  not  put  in  till  the  1st  of  May; 
what  sum  did  he  then  put  in,  to  have  an  equal  share  with  A.  at 
the  expiration  of  the  year?  Ans.  D1500. 

21.  A.,  B.,  and  C,  entered  into  partnership  for  12  months; 
A.  put  in  at  first  D873.60,  and  4  months  after  he  put  in  D96 
more ;  B.  put  in  at  first  D979.20,  and  at  the  end  of  7  months 
he  took  out  D206.40 ;  C.  put  in  at  first  D355.20,  and  3  months 
after  he  put  in  D206.40,  and  5  months  after  that  he  put  in 
D240  more  ;  at  the  end  of  12  months  their  gain  is  D3446.40; 
what  is  each  man's  share  of  the  gain  ? 

Ans.  A.  D1334.82.5,  B.  D1271,61.4,  C.  D839.96. 
16 


182 


PARTNERSHIP. 


22.  Two  merchants  entered  into  company  for  18  months  ;  A. 
at  first  put  in  D500,  and  at  the  end  of  8  months  he  put  in  DlOO 
more  :  B.  at  first  put  in  D800,  and  at  the  end  of  4  months  he  took 
out  D200  ;  at  the  expiration  of  the  time  they  find  their  gain  to 
be  D700  ;  what  is  each  man's  share  of  the  gain  ? 

Ans.  A.  D324.07.4,  B.  D375.92.5+ 

23.  Three  persons,  A.,  B.,  and  C,  made  a  stock  for  12 
months ;  A.  put  in  at  first  D580,  3  months  after  he  put  in  DlOO^ 
more  ;  B.  put  in  at  first  DIOOO,  and  after  9  months  he  put  in 
D200  ;  C.  put  in  at  first  D486,  3  months  after  he  took  out  D300, 
and  2  months  after  he  put  in  D500  ;  3  months  after  this  he  took 
out  D400,  and  1  month  after  he  put  in  DlOOO ;  at  the  end  of  12 
months  their  gain  was  D2108.44  ;  what  is  each  man's  share "? 

A.  D580x3=l740 
680x9=6020  + 


7860A. 


B.  DIOOO  X  9:^9000   1 

200  + 


C.  D486x3  =  1458 
-300 

+  186X2X      372 
500 

686x3=   2058 
—400 


1200X3  =  3600     I      +286x1=     286 

j         1000 

12600B.  J 


1286x3=  3858 


7860  A. 

12600  B. 

8032  C. 


28492  div. 


;>A.581.64.8+ 
B.932.41.4 
C.594.37.7 


2108.44.0Pr. 


8032C. 

24.  A  gentleman  left  an  estate  at  his  death  of  D30000,  to  be 
divided  among  his  5  children,  in  such  a  manner  that  their 
shares  should  be  to  each  other  as  their  ages,  which  are  7,  10, 
12,  15,  16  ;  required  the  share  of  each. 

Ans.  D3500,  5000,  6000,  75000,  8000. 

25.  A.  and  B.  entered  into  partnership  for  16  months  ;  A.  put 
in  D1200  at  first,  and  9  months  after  D200  more ;  B.  put  in  at 
first  D1500,  at  the  end  of  6  months  he  took  out  D500 ;  their 
gain  was  D772.20  ;  what  is  the  share  of  each  ? 

Ans.  A.'s  D401.70,  B.'s  D370.50. 
Note. — As  an  evidence  of  the  correctness  of  the  rules  of  part- 
nership, and  that  their  loss  or  gain  is  in  proportion  to  their 
stocks  in  trade,  let  A.'s  stock  be  200,  and  B.'s  DlOO,  and  their 
OSS  or  gain  D37.50,  which  is  at  the  rate  of  12j  per  cent. ;  A. 
will  gain  or  lose  D25,and  B.  D12. 50,  because  A.'s  stock  is  just 


PERCENTAGE.  183 

• 

twice  as  much  as  B.'s,  consequently  his  loss  or  gain  must  be 
twice  as  much  ;  and  the  same  principle  will  hold  good  in  partner- 
ship with  or  without  time,  that  is,  the  gain  or  loss  must  be  in 
proportion  to  the  stock,  and  time  that  the  partnership  continued. 
If  A.  and  B.  enter  into  partnership  for  one  year,  and  A.  puts  in 
D200,  and  B.  DlOO,  but  at  the  expiration  of  6  months,  A.  finds 
it  convenient  to  withdraw  his  D200,  and  the  partnership  con- 
tinues to  the  end  of  the  year,  and  their  gain  is  D37.50,  it  must 
be  divided  equally.  Again  :  If  A.  commence  trade  with  a  cap- 
ital of  D500,  and  at  the  expiration  of  6  months  he  shall  receive 
B.  into  partnership  with  a  capital  of  DlOOO,  their  loss  or  gain  at 
the  end  of  12  months  will  be  equal ;  if  at  6  per  cent.  (D60)  it 
will  be  D30  each. 


What  is  partnership  ?  When  is  it  used  ?  When  time  is  not 
considered,  what  is  the  rule  ?  How  will  you  prepare  the  ques- 
tion for  solution  ?  What  can  you  say  of  the  assessment  of  taxes  ? 
What  is  the  first  thing  to  be  done  ?  How  will  you  proceed  to 
find  the  amount  of  taxes  to  be  paid  by  A.,  B.,  C,  &c.  ?  When 
time  is  considered  in  partnership,  what  is  the  rule  ?  Repeat  the 
2d  and  4th  rules.  Method  of  proof.  What  is  capital  or  stock  ? 
On  what  principles  are  the  rules  of  partnership  founded  ? 

26.  E.,  F.,  and  G.,  formed  a  partnership  ;  E.  put  in  D400for 
.75  of  a  year,  F.  D300  for  .5  of  a  year,  and  G.  D500  for  .25 
of  a  year,  with  which  they  gained  D720 ;  required  the  share 
of  each. 

Ans.  E.  D375l|  ;  F.  Dl87jf  ;  G.  Dl56i|=D720  proof. 

27.  A.  put  in  ^  for  |  of  a  year,  B.  f  for  ^  a  year,  and  C.  the 
rest  for  1  year  ;  their  joint  stock  was  1,  and  their  gain  1  ;  what 
is  each  man's  share  1  Ans.  A.'s  is  i| ;  B.'s  ^j  ;  C.'s  ^^y=||  ~1 


PERCENTAGE. 

When  we  speak  of  per  cent.,  it  generally  has  reference  to 
interest  or  discount  on  money,  and  means  the  hundredth  part  of 
the  thing  spoken  of ;  for  we  can  say  so  much  per  cent,  of  a 
bushel,  yard,  &c.,  as  well  as  money  ;  when  we  say  10  per  cent., 
we  mean  DlO  on  the  hundred,  or  the  10th  part  of  100 ;  as  B. 
spends  10  per  ct.  of  his  DlOO,  he  would  have  but  D90  remaining 


184  PERCENTAOE. 


EXAMPLES. 


1.  What  is  the  difference  between  D500  at  7.5  per  cent.,  and 
D500  at  8  per  cent.  ? 

Thus:  D500x7.5=D37.50;  D500x8=D40.00;  difference 
2.50.     Ans. 

2.  Two  men  had  each  D240 ;  one  of  them  spends  14  per 
cent.,  and  the  other  18^  per  cent.  ;  how  much  more  did  one 
spend  than  the  other  ? 

D240X  14  =  D33.60  ;  D240+18^  X  D44.40  ;  difference 
DlO.80.     Ans. 

To  find  the  rate  per  cent. 

RULE    I. 

1.  Bring  the  number  to  hundreds  by  annexing  two  ciphers,  oi 
removing  the  decimal  point  two  places  to  the  right. 

2.  Divide  the  numbers  so  formed  by  the  sum  on  which  the 
percentage  is  estimated  ;  the  quotient  will  express  the  per  cent. 

3.  A  merchant  goes  to  New  York  with  D1500 ;  he  first  lays 
out  20  per  cent.,  after  which  he  expends  D660 ;  what  per  cent, 
was  his  last  purchase  of  the  money  that  remained  after  his  first  ? 
Thus:  D1500X20  per  ct.=zD300:  1500  —  300=1200)66000(55 
per  cent.     Ans. 

4.  If  I  pay  D679.84  for  750  bushels  of  wheat,  and  sell  the 
same  for  1)874.50 ;  how  much  do  I  make  per  cent,  on  what  I 
paid,  and  on  the  sum  received  ? 

5.  If  I  contract  a  debt  of  D500  and  make  a  payment  of 
D350,  what  per  cent,  of  the  debt  do  I  pay  ? 

When  the  per  cent,  of  loss  or  gain  is  given,  and  the  amount  re- 
ceived, to  find  the  principal  cost. 

6.  I  sell  a  quantity  of  goods  for  D170,  by  which  I  lose  15 
per  cent. ;  what  did  they  cost  1 

Aw^.DlOO  — 15  =  85)170x100(200  D.  cost. 

7.  Sold  goods  to  the  amount  of  D225,  and  made  20  per 
cent. ;  what  did  they  cost  ? 

To  find  the  percentage  on  lands,  or  allowance  for  roads,  Sfc. 

It  is  customary  in  Pennsylvania,  and  probably  in  many  other 
states,  to  deduct  6  acres  out  of  106,  for  roads,  &;c.  ;  the  land 
before  the  deduction  is  made  may  be  termed  the  gross,  and  that 
remaining  after  each  deduction,  the  net  or  strict  measure. 


GROSS,    TARE,    AND    NET    WEIGHT.  185 


teauce  the  gross  to  perches,  and  divide  by  1.06,  and  the  quo- 
tient will  be  the  answer  in  perches,  strict  measure. 

Multiply  the  net  or  strict  measure  by  1.06,  and  the  product 
will  give  the  gross  measure  or  quantity ;  or  work  decimally. 

EXAMPLES. 

1 .  How  much  net  land  or  strict  measure  is  there  in  a  tract 
:)f  901  A.  2  R.  26  po.  gross? 

Thus,  901  A.  2  R.  26  po. =901.6625-^  1.06  =  850.625  A.= 
850  A.,  2  R.,  20  po.     Ans. 

2/*  How  much  land  must  I  enclose  to  have  850  A.  2  R.  20  po., 
net?  Thus  850.625x1.06=901.6625  A.=901  A.2R.26po., 
gross.     Ans. 

Note. — These  two  operations  prove  each  other. 


RULE    IT. 


^^B  Divide  the  content  in  perches  by  169.6,  which  will  give  the 
'^ret  in  all  cases,  where  the  given  quantity  is  106  A.,  and  ratio  6. 

Thus  :  901  A.  2  R.  26  po.  =  144266  po.-f- 169.6  =  850.625  A. 
=  850  A.  2  R.  20  po.     Ans. 

Or:  850.625x169.6  =  144266  po.=901  A.  2  R.  26  po.  as 
above. 

Again:  106  A.  =  16960po.~169.6=100  A. ;  or  16960-7-1.06 
=  16000po.  =  100  A. 

Note. — The  deduction  to  be  made  on  every  106  A.  is  9.6 
perches  ;  this  added  to  160  po.=  169.6  po.,  hence  the  divisor. 


GROSS,  TARE,  AND  NET  WEIGHT. 

The  following  questions  are  usually  denominated  under  the 
rule  or  appellation  of  Tare  and  Tret,  The  use  and  application 
of  the  several  rules  are  for  deducting  certain  allowances  which 
are  made  by  merchants  and  tradesmen  in  selling  their  goods  by 
weight,  for  the  purpose  of  making  the  proper  deduction  to  ascer- 
tain the  net  weight.  In  England  these  rules  are  in  constant 
use,  but  a  better  system  is  being  introduced  into  this  country, 
that  of  taking  100  pounds  as  the  true  weight,  in  place  of  112  lb. 
gross  ;  then  all  that  will  be  required  will  be  to  ascertain  the 
weight  of  the  box,  cask,  bag,  &c.,  containing  the  article,  and 

16* 


186  GROSS,    TARE,    AND    NET    WEIGHT, 

the  remainder  will  be  the  net  or  true  weight.  Tlie  collections 
at  the  customhouse,  or  United  States  duties,  are  connected  with 
gross  weight. 

Gross  weight  is  the  whole  weight  of  the  goods,  together  with 
the  weight  of  the  cask,  bag,  &c.,  which  contains  them. 

Tare  is  an  allowance  made  to  the  buyer,  or  the  deduction  of 
the  weight  of  the  cask,  box,  bag,  (fee,  containing  the  articles 
sold. 

Net  weight  is  what  remains  after  all  the  deductions  are  made 

Note. — The  following  questions  are  only  the  application  of 
the  rules  of  proportion  and  practice. 

When  the  tare  is  so  much  on  a  given  quantity » 

RULE    I. 

Subtract  the  given  tare  from  the  given  quantity,  and  the  re 
mainder  will  be  the  net  weight. 

1 .  What  is  the  net  weight  of  a  cask  of  sugar,  weighing  7  cwt 
1  qr.  16  lb.  gross,  tare  3  qrs.  18  lb.  ? 

7  cwt.  1  qr.  16  lb. 
-  3         18 


6  1  26  Ans. 

2.  What  is  the  net  weight  of  a  cask  of  rice,  weighing  5  cwt. 
gross  ;  tare  2  qrs.  13  lb.  ?  Ans.  4  cwt.  1  qr.  15  lb. 

3.  Required  the  net  weight  of  a  hogshead  of  sugar,  weighing 
gross,  8  cwt.  1  qr.  22  lb.  ;  tare  3  qrs.  9  lb.  ? 

Ans.  7  cwt.  2  qrs.  13  lb. 

4.  What  is  the  net  weight  of  175  cwt.  2  qrs.  20  lb. ;  tare 
6  cwt.  2  qrs.  25  lb.  ?  Ans.  168  cwt.  3  qrs.  23  lb. 

5.  What  is  the  net  weight  of  4  casks  of  sugar,  each  weigh- 
ing 5  cwt.  2  qrs.  12  lb.  gross ;  tare  per  cask,  2  qrs.  18  lb. 

Ans.  19  cwt.  3  qrs.  4  lb. 

When  the  tare  is  so  much  per  cask,  box,  bag,  <SfC. 

RULE    II. 

Multiply  the  given  tare  per  bag,  box,  &c.,  by  the  number  of 
bags,  boxes,  &c.,  and  subtract  the  product  from  the  gross  weight, 
and  the  remainder  is  the  net  weight. 

6.  A.  sold  5  casks  of  rice,  which  weighed  1137  lbs.  each,  and 
each  cask  weighed  75  lbs.  ;  required  the  net  weight  and  value 
of  the  tobacco  at  14  cents  per  pound  ? 

Ans.  Net  weight,  5310  lb.  ;  value,  D743.40. 


UNITED  STATES  DUTIES.  187 

7.  Received  per  ship  Napoleon,  from  South  Atneiica,  55  bags 
of  coffee,  the  gross  weight  of  which  is  2201bs.  each,  and  the 
weight  of  each  sack  is  olbs. ;  required  the  net  weight  and  value 
of  the  coffee  at  16  cents  per  pound  ? 

A?is.  net  weight  11825lbs.;  value  D1892. 

8.  Received  from  Salina  72  bags  of  salt,  the  weight  of  each 
bag  is  2101b.  gross,  and  each  sack  weighs  71bs. ;  what  is  the 
net  weight,  and  what  does  it  come  to  at  4  cts.  per  pound  ? 

Ans.  weight  146161bs. ;  value  D584.64. 

9.  Bou^  it  1 10  hogsheads  of  sugar,  gross  723lbs.  each,  weight 
of  each  hogshead  62  lbs. ;  what  is  the  net  weight,  and  what 
will  it  come  to  at  12.5  cts.  per  pound  ? 

Ans.  727101bs.;  amount  D9088.75. 

10.  Bought  5  casks  of  rice,  which  weighed  gross  18  cwt.,  2 
qrs.,  12  lbs.,  tare  per  cask  45  lbs.,  for  which  I  paid  D5.50  per 
cwt.,  and  sold  the  same  for  D6.25  per  cwt. ;  required  the  net 
weight,  cost,  amount  it  sold  for,  and  profit  ? 

Ans.  net  weight  1859  lbs. ;  cost  D91.29  ;  sold  for  D103.74  ; 
profit  Di2.45. 

REVIEW. 

What  is  gross  weight  ?  What  is  tare  ?  What  is  net  weight  ? 
Repeat  rule  Ist.     Repeat  rule  2d. 


UNITED  STATES  DUTIES. 

In  all  civilized  countries,  where  merchandise  or  goods  are 
imported,  importers  are  required  to  pay  a  certain  amount  of  their 
value,  at  a  certain  rate  per  pound,  hundred,  yard,  or  gallon  ;  this 
is  called  duty,  which  is  established  and  collected  by  the  laws 
of  the  country  where  the  goods  are  landed  ;  for  this  purpose, 
customhouses  are  erected  in  the  seaport  towns  to  collect  the 
custom  or  duties,  tonnage  of  vessels,  port  duties,  &c.,  which  to- 
gether are  called  the  revenue.  An  ad-valorem  duty  is  such  a 
per  cent,  on  the  actual  cost  of  the  goods  in  the  country  from 
which  they  are  imported  ;  thus  an  ad-valorem  duty  of  20  per 
cent,  on  tea  from  China,  is  a  duty  of  20  per  cent,  on  the  cost  of 
the  tea  in  China ;  the  duties  are  computed  on  the  net  weight. 

EXAMPLES. 

1.  What  is  the  duty  on  1400  lbs.  of  coffee  at  2  J-  cts.  per  lb. "? 

Ans.  1400x2^=D3.50 


188  SINGLE    POSITION. 

2.  If  the  duty  on  molasses  is  5  cents  on  a  gallon  when  im 
ported  in  an  American  vessel,  and  10  per  cent,  more  in  a  foreigii 
vessel ;  what  is  the  duty  on  3950  gallons  in  both  vessels  ? 

Ans.  D197.50  and  D217.25 

3.  What    is    the    duty   on    goods    which   cost    in    Calcutta 
D2780.50,  at  12^  per  cent,  ad  valorem?  Ans.  D347.561 


POSITION. 

Position  is  a  rule  founded  on  the  principles  of  proportion, 
and  by  working  with  one  or  more  supposed  numbers,  as  real 
numbers,  we  can  discover  the  true  number  or  answer  to  the 
question.  It  is  of  two  kinds,  termed  single  and  double.  Single 
Position  is  when  only  one  supposed  number  is  necessary  for 
the  operation.  Such  questions  as  are  usually  given  in  arith- 
metic for  solution,  by  working  with  supposed  (improperly  called 
false)  numbers,  are  equations  in  algebra,  by  which  they  are 
more  conveniently  and  readily  solved  by  those  who  are  ac- 
quainted with  that  science.  (For  an  illustration  of  the  rule,  see 
Double  Position.) 


SINGLE  POSITION. 

RULE. 

Suppose  a  number,  and  work  with  it  as  though  it  was  the 
true  number,  according  to  the  nature  of  the  question  ;  then,  as 
the  result  of  that  operation  :  is  to  the  given  number  : :  so  is  the 
number  supposed  :  to  the  number  required.  Proof,  add  the 
several  results  together. 

QUESTION. 

1 .  A.  owes  a  certain  sum  of  money :  |,  g^,  },  is  D500  ;  what 
is  the  amount  of  the  debt  ? 

Suppose  debt  D1200  ^=300  ^  then,  650  :  500  ::  1200 
-1  =  150  >  1200 


^=200  )  D 


650)600000(923.07.7  A 

Sum  of  shares,     D650 


DOUBLE    POSITION.  18^ 

2.  A  certain  sum  of  money  is  given  to  4  persons  :   to  A.  J, 
B.  \,  C.  -g-,  and  D.  draws  the  rest,  which  is  D28  ;  required  th^ 
sum.     (This  question  can  be  solved  by  Double  Position.) 
Suppose  60,  iz:z20  A.  ]  Then,    15  :  28  ::  60  :  112.  Ans. 
45  i=15  B.    I  Proof,  112,  J=37.33^  A. 
—  i=10C.   ;>  112,1=28.00     B. 


ffference,  15        — 
Sum  of  parts,  45 


112,  ^  =  18.66f  C. 
D.  took,     28.00     D. 


D112.00 
f  3.  What  sum  is  that  of  which  ^,  -J,  and  ^,  make  D94  ? 

Ans.  D120. 

^4.  A  person  said  that  ^,  ^,  J,  and  i  of  the  money  in  his  pos- 

kssion  would  make  D57  ;  how  much  had  he  ?  Ans.  D60. 

I5.  The  ages  of  A.,  B.,  and  C,  amount  to  133  years ;  B.  is  J 

cler  than  C,  and  A.  is  J  older  than  B. ;  required  their  separate 

fires.  Ans.  A.  56  yrs.,  B.  42  yrs.,  C.  35  yrs. 

6.  What  sum  of  money  at  6  per  cent,  per  annum,  simple  in« 

terest,  will  amount  to  D500  in  10  years  1  Ans.  D312.50. 

!7B.'s  age  is  1.5  of  A.'s,  C.'s  twice  as  much  as  B.'s,  and  their 

united  ages  make  ]  32  years  ;  required  the  age  of  each  1 

Ans.  A.  24,  B.  36,  C.  72  years. 

8.  WKM  is  the  age  of  a  person  who  says  that  if  -f^  of  the 
years  he  has  lived  be  multiplied  by  7,  and  f  of  them  be  added 
to  the  product,  the  sum  would  be  292  1  Ans.  60  years. 

9.  A  bankrupt  is  indebted  to  A.  D284.60,  to  B.  D794.18,  to 
C.  D651.44,  and  to  D.  D491.25,  and  his  estate  is  worth 
D1460.5;  if  the  whole  were  divided,  how  much  would  each 
creditor  receive  l^Z Yf^-  i     "  ^C^ S-  b  @*  1    (^  ir^  l)  )  ^ 

DOUBLE  POSITION 

Teaches  to  solve  such  questions  as  require  two  supposed  numbers 
in  the  operation, 

RULE    I. 

Suppose  two  numbers,  and  work  with  each  agreeably  to  the 
tenor  of  the  question,  observing  the  errors  of  the  results  ;  multi- 
ply the  errors  of  each  operation  into  the  supposed  number  of  the 
othw ;  then,  if  the  errors  be  alike  (that  is,  both  too  much,  or  both 


190  DOUBLE    POSITION. 

too  small),  take  their  difference  for  a  divisor,  and  the  difference 
of  the  products  for  a  dividend ;  but  if  not  alike,  then  take  their 
sum  for  a  divisor,  and  the  sum  of  the  products  for  a  dividend. 

QUESTIONS. 

1.  A.,  B.,  and  C,  agree  to  pay  a  debt  of  D500,  of  which  A.  is 
to  pay  a  certain  sum  ;  B.  is  to  pay  DlO  more  than  A.,  and  C.  is  to 
pay  as  much  as  both  A.  and  B. ;  how  much  must  each  one  pay  ? 
Suppose  1st,  A.    90     Suppose  2d,  A.  100       500       500 

B.  100  B.  110  —380  —420 

C.  190  C.  210       

120  err.  80  error 

Sum     380  420     —80 

40  Diff.  error. 
Then,  120  error,  xlOO  A=. 12000  120  A. 

80  error,  X   90  A.=  7200  130B.}Ans. 


250  C 


:l 


Difference  of  error,      40)4800(120  ^w^.   

500  proof. 
Note. — This  rule  is  founded  on  the  supposition  that  the  first 
error  is  to  the  second,  as  the  difference  between  the  true  and 
supposed  number  is  to  the  difference  between  the  true  and  sec- 
ond supposed  number.  When  that  is  not  the  case,  the  exact 
answer  to  the  question  can  not  be  found  by  this  rule. 

KULE    II. 

1.  ''Take  any*two  convenient  numbers  and  proceed  with  each 
according  to  the  conditions^of  the  question.  2.  Place  the  result 
or  errors  against  their  positions  or  supposed  numbers,  thus  "^  ^  X  ^  ^ 
and  if  the  error  be  too  great,  mark  it  with  +  ;  if  too  small,  with 
— .  3.  Multiply  them  crosswise  ;  that  is,  the  first  position  by 
the  last  error,  and  the  last  position  by  the  first  error.  4.  If  the 
errors  be  alike,  that  is,  both  too  small  or  both  too  great,  divide 
the  difference  of  the  products  by  the  difference  of  the  errors, 
and  the  quotient  will  be  the  answer.  5.  If  the  errors  be  unlike, 
that  is,  one  too  small  and  the  other  too  great,  divide  the  sum  of 
the  products  by  the  sum  of  the  errors,  and  the  quotient  will  be 
the  ariswer. 

Note, — When  the  errors  are  the  same  in  quantity,  and  unlike 
in  quality,  half  the  sum  of  the  supposition  is  the  number  sought 
Observe  the  following  examples  and  explanation. 


DOUBLE    POSITION. 


19] 


500 
420 

2  er.  80 


EXAMPLE  2.  (1st  ques.) 

Supposition  1.  A.    90  Sup.  2.  A.  100  given  num.  500 
B.  100  B.  110  am.  of  sup.  380 

IK  C.  190  C.210  

V^m  1  er'r  120 

H[  Am't  of  sup.— 380  Am.  of  sup.  420 

l^lfer.  120")  Istsup.    90    120 Ister.  f  then,    12000 

2^"     ^n2dsup.l00^-802der.^  "J^ 

Dif.er.40j  120  x    90  x       (^  d.er.40)4800(120D.A.pM 

12000  7200  130      B.  " 

250      C.  »' 


500D.  proof. 


EXAMPLE    III. 


then  the  errors  are  unlike^  that  is,  one  plus  and  the  other  minus ^ 


1st,  A.  100  2d,  A.  140- 

B.  110  B.  150 

C.  210  C.  290  y 

I   minus 

420  580  J 

80  100x80=  8000 
80  ■^140x80  =  11200"^ 


500 
•420 


580 
—500 


80-1  er.  SO  +  plus 
(*see  remarks.) 
120  A. 
130  B. 
250  C. 


160 


160)19200(120  A. 

500  proof, 

2.  A.  is  20  years  of  age ;  B.'s  age  is  A.'s,  and  half  of  C.'s; 
and  C's  is  equal  to  them  both  ;  their  several  ages  are  required. 

Ans.  A.  20,  B.  60,  C.  80. 

3.  Three  persons  pay  the  sum  of  DlOO  ;  B.  paid  DlO  more 
than  A.,  and  C.  as  much  as  A.  and  B. ;  how  much  did  each  pay  ? 

Ans.  A.  20,  B.  30,  C.  50. 

4.  What  sum  is  that,  which  being  increased  by  its  \,  its  \, 
and  18  more,  will  be  doubled?  Ans.  72. 

5.  A.,  B.,  and  C,  receive  D324,  but  not  agreeing  in  the  divis- 
ion of  ir,  each  took  as  many  as  he  could  :  A  got  a  certain  num- 
ber;  B.  as  many  as  A.  and  Dl5  more;  C.  got  a  fifth  part  of 
the  sums  added  together ;  how  much  did  each  get  ? 

Ans.  A.  D127.50,  B.  D142.50,  C.  D54. 

6.  Divide  DlOO,  so  that^B.  may  have  twice  as  much  as  A., 
wanting  D8  ;  and  C.  three' times  as  much,  wanting  D15  ;  what 
is  the  share  of  each  ?        *Ans.  A.  D20.50,  B.  D33,  C.  D46.50. 


192  DOUBLE    POSITfON. 

7.  A  certain  sum  of  money  is  given  to  four  persons :  to  A 
•J,  B.  i,  and  C.  |,  and  D.  draws  the  rest,  which  is  D28  ;  re 
quired  the  sum.     (See  question  2,  Single  Position.) 


Sup.  1, 

H  22.50        i,  24.75  2.25  difference  of  error. 

1 

J).]  28  D".,  28  99x5.5   =544.5 


First  operation. 

90     Sup.  2, 

99                 5.5 

— 

—                 3.25 

30              i, 

33                  

22.50        I, 

24.75              2.25  i 

15              h 

16.50 

90x3.25=292.5— 


95.50  102.25  

—90.00         —99.00  2.25)252.0(112    Ans, 


5.5  error.      3.25-  error. 

Second  operation. 
Sup.  1,90     Sup.  2,  60  13      error. 


3' 

30 

20 

T' 

22.5 

15 

"B» 

15 

10 

\ 

28 

28 

95.5 

73 

90 

—60 

5.5  error. 

7.5  difference  of  errors 

90x13  =1170 
60X5.5=  330— 


7.5)840(112  ^n^'. 

5.5  error.     13  error. 
Ih  the  above  case  the  suppositions  are  varied  to  illustrate  the 
rule,  but  in  both  statements  the  errors  are  alike,  that  is,  both  too 
small,  or  minus.     In  the  following  example  the  errors  are  un- 
like, that  is,  one  is  minus  and  the  other  pkis. 


Third  operation. 

Sup.  1, 

99     Sup 

.2,  120 

120          102.25         3.25 

— 

118           99              2 

h 

33 

h  ^0 

i^ 

24.75 

h  30 

2  er.       3.25  er.    5.25  sum.  er. 

h 

16.50 

h  20 

D., 

28 

D.,  28 

99x2      =198 
130x3.25=390^ 

102.25  118  

'  5.25)588(112  Ans. 


BOUBLE    POSITION.  193 

REMARKS. 

In  the  above  examples,  I  find  the  errors  to  be  alike --that  is. 
both  too  small;  because  the  sum  in  either  case  is  less  than 
D500  the  given  number  ;  therefore,  after  proceeding  according 
to  rule,  till  I  find  the  difference  of  the  errors,  and  then  make  the 
statement  X  for  obtaining  the  answers,  you  will  observe  that  the 
statement  or  terms  are  muhiplied  crosswise  ;  then  take  the  dif 
ference  of  the  products  and  divide  it  by  the  difference  of  th 
errors,  which  is  40,  and  yoti  get  the  first  or  A.'s  answer ;  from 
this  you  can  easily  get  the  others.  Had  the  errors  been  unlike, 
that  is,  one  too  great  and  the  other  too  small,  you  would  divide 
the  sum  of  the  products  by  the  sum  of  the  errors,  and  the  quo- 
tient would  be  the  answer.  It  is  immaterial  what  numbers  are 
used  for  the  suppositions,  provided  you  observe  the  principles 
of  the  rule  and  the  examples  above,  for  the  result  will  always 
be  the  same.  As  has  been  before  observed,  this  rule  has  its 
origin  in  proportion ;  the  reason  for  changing  the  errors  or  sup- 
positions to  obtain  a  different  result  or  answer,  is  accounted  for 
on  the  same  principles  that  the  1st  and  ^d  terms  in  Proportion 
are  varied,  or  changed,  to  gain  the  required  result ;  in  that  rule 
we  change  the  first  and  second  terms ;  if  the  answer  is  to  be 
less,  then  the  greater  of  the  two  terms  is  the  divisor  j  if  the 
answer  is  to  be  more,  then  the  less  of  the  two  terms  is  the 
divisor,  &c. 

Algebraical  deinonstration, — Let  A  and  B  be  two  numbers, 
produced  from  a  and  h  by  similar  operations ;  it  is  required  to 
find  the  number  from  which  N  is  produced  by  a  like  operation. 
Put  a:=number  required,  and  let  N  — A=r,  and  N  — Br=5. 
Then,  according  to  the  supposition  on  which  the  rule  is  found- 
ed, r  :  s  ::  x—a  :  x—b;  whence,  by  multiplying  means  and 
extremes,  rx—rb=sx—sa;  and  by  transposition,  rx~sx=rb  — 
sa ;  and  by  division,  rb~as 

x=-r-. — — r=the  number  sought ;  and  if  r  and 
r — s 
s  be  both  negative,  the  theorem  is  the  same  ;   and  if  r  or  ,y  be 

rb-^-sa 

negative,  x  will  be  equal  to  —■ which  is  the  rule. 

r  -\-   s 

'  REVIEW. 

What  is  Position  ?  How  many  kinds  are  there  ?  What  is 
Single  Position  ?  What  is  the  rule  ?  What  is  Double  Position  T 
What  is  first  to  be  done  when  you  commence  an  operation  iu 

17 


194  INVOLUTION'    OR    THE    RAISING    OF    POWERS. 

Single  Position  ?  After  having  ascertained  the  result  of  the 
operation,  how  will  you  proceed?  How  will  you  first  begin  an 
operation  in  Double  Position  ?  After  having  obtained  the  first 
error,  how  will  you  proceed?  When  you  have  obtained  the 
second  error,  what  is  to  be  done  ?  What  will  yon  do  after  you 
have  multiplied  the  second  supposition  by  the  first  error,  and 
the  first  supposition  by  the  second  error?  When  you  have 
ascertained  whether  the  errors  are  both  of  the  same  kind,  how 
do  you  proceed  ?  If  they  are  not  of  the  same  kind,  how  will 
you  proceed  ?     What  is  the  rule  fir  Double  Position  ? 

8.  A  laborer  was  hired  for  60  days  upon  this  condition  :  that 
for  every  day  he  wrought,  he  should  receive  75  cents,  and  for 
every  day  he  was  idle,  he  should  forfeit  37i  cents  ;  at  the  ex- 
piration of  the  time  he  received  Dl8  ;  how  many  days  did  he 
work,  and  how  many 'W.%  ha  idle  ? 

^        Ans.  worked  36  days,  was  idle  24  days. 

9  Two  persons,  A.^apd  B.,  have  the  same  income  ;  A.  saves 
}  of  his  yearly  ;  but  B?V^b^ spending  D150  per  annum  more  than 
A.,  at  the  end  of  8  years *finds  himself  D400  in  debt:;  what  is 
their  income,  and  what  does  each  spend  per  annum  ? 

Ans,  income  D400 ;  A  spends  D300  •  B.  D450. 


INVOLUTION,  OR  THE  RAISING   OF  POWERS, 
Teaches  the  method  of  finding  the  powers  of  numbers, 

A  POWER  is  the  product  arising  from  multiplying  any  given 
number  into  itself  continually  a  certain  number  of  times  ;  thus, 
2x2=4,  is  the  second  power  or  square  of  2;  2x2x2=:  8,  the 
third  power  or  cube  of  2  ;  2x2x2x2  =  16,  the  fourth  power 
of  2,  &LC.  The  mimber  whicl;^  denotes  a  power  is  called  its  in- 
dex. If  two  or  more  powers  are  multiplied  together,  their  prod- 
uct is  that  power  whose  index  is  the  sum  of  the  exponents  of 
the  factor  ;  thus  2x2=4,  square  of  2  ;  4x4=16,  fourth  power 
of  2;  and  16x16=256,  eighth  power  of  2,  &c.  The  power 
often  denoted  by  a  figure  placed  at  the  right  and  a  little  abovt) 
-he  number,  which  figure  is  called  the  index  or  exponent  of  that 
power  (thus,  2^,  3^),  and  is  always  one  more  than  the  number 
of  multiplications  to  produce  the  power,  or  is  equal  to  the  num- 


INYOLUTION,   OK   TWE   RAISING   OF   POWERS.  195 

ber  of  times  the  given  number  is  used  as  a  factor  in  producinig 
the  power.  In  producing  the  square  of  2,  there  is  only  one  mul- 
tiplication, or  two  factors ;  in  producing  the  cube,  there  are 
two  multiplications  or  three  factors,  3^  x 3x3=27,  &c.  This 
subject  will  be  more  fully  illustrated  in  progression. 


Multiply  the  given  number  or  first  power  continually  by  itself 
till  the  number  of  multiplications  be  1  less  than  the  index  of  the 
power  to  be  found,  and  the  last  product  will  be  the  power  re- 
quired. Fractions  are  multiplied  by  taking  the  products  of  their 
numerators,  and  of  their  denominators;  they  will  be  involved 
by  raising  each  of  their  terms  to  the  power  required,  and  if  a 
mixed  number  be  proposed,  either  reduce  it  to  an  improper  frac- 
tion, or  reduce  the  vulgar  fraction  to  a  decimal,  and  proceed  by 
the  rule. 

EXAMPLES. 

1 .  Required  the  third  power  or  cube  of  35. 

Ans.  35  X  35  X  35=42875. 

2.  What  is  the  5th  power  of  7  ?  Ans.  16807. 

3.  What  is  the  5th  power  of  9  ?  Ans.  59049=9^. 

4.  What  is  the  5th  power  of  f  ?  Ans,  -^yts' 

5.  What  is  the  4th  power  of  .045  ? 

Ans.   .000004100625. 

Here  we  see  that  in  raising  a  fraction  to  a  higher  power,  we  de^ 
crease  its  value. 

6.  What  is  the  third  power  of  .263  ?  Ans.  .018191447. 

7.  What  is  the  eighth  power  of  i  ?  Ans.  -g-^j^^. 

8.  What  is  the  square  of  60  ?  Ans.  3600. 

9.  What  is  the  square  of  i  ?  Ans.  i. 

10.  What  is  the  square  of  .01  1  Ans.  .0001. 

11.  What  is  the  cube  of  2i  ?  Ans.  ll||. 

12.  What  is  the  ninth  power  of  747  ? 

13.  What  is  the  seventh  power  of  298.75  ? 

REVIEW. 

What  is  a  power  ?  How  do  you  raise  a  number  to  any  pow- 
er ?    What  is  the  rule  ? 


196  .  SQUARE  ROOT. 

EVOLUTION,  OR  THE  EXTRACTION  OF  ROOTS. 

Evolution,  or  the  extraction  of  roots,  properly  belongs  to 
mathematics,  and  without  a  knowledge  of  that  science,  it  will 
require  strict  attention  and  close  application  to  arrive  at  any 
degree  of  perfection  in  the  use  and  principles  of  those  rules. 
The  most  correct  and  convenient  method  of  extracting  the  roots 
of  the  several  powers,  particularly  those  of  the  higher  order,  is 
by  logarithmic  tables,  as  far  preferable  to  any  rules  that  can  be 
given  in  common  arithmetic. 

The  root  of  a  number,  or  power,  is  such  a  number  as,  being 
multiplied  into  itself  a  certain  number  of  times,  will  produce  that 
number  or  power,  and  is  denominated  the  square,  cube,  biquad- 
rate,  &:c.,  or  2d,  3d,  and  4th  root,  accordingly  as  it  is,  when 
raised  to  the  2d,  3d,  and  4th  power,  equal  to  that  power.  Thus 
4  is  the  square  root  of  16,  because  4x4  =  16  :  and  4  is  the  cube 
root  of  64,  because  4x4x4=64  :  and  4  is  the  fourth  root  or 
biquadrate  of  256,  because  4  x4  x4  X4=:256,  &c.  The  roots 
are  proportional,  but  their  proportion  is  different  from  simple  or 
compound  proportion;  the  raising  of  powers  increase  in  a 
uniform  ratio,  but  this  will  not  always — indeed  but  seldom — 
occur  in  the  extraction  of  roots.  Although  there  is  no  number 
of  which  we  can  not  find  any  power  exactly,  yet  there  are 
many  numbers  of  which  precise  or  exact  roots  can  never  be 
determined ;  but  by  the  use  of  decimals  we  can  approximate 
toward  the  root  to  any  assigned  degree  of  accuracy ;  those 
roots  are  called  surds  ;  and  those  which  are  perfectly  accurate, 
rational  roots ;  surd  roots  sometimes  have  tbeir  origin  in  circu- 
lating decimals,  or  vulgar  fractions.  As  few  numbers  are  com- 
plete powers,  surds  must  very  often  occur  in  arithmetical  opera- 
tions, but  the  result  can  be  obtained  nearly  by  continuing  the 
extraction  of  the  root. 


SQUARE  ROOT. 

RULE    I. 


] .  Separate  the  given  number  into  periods  of  two  figures 
each,  beginning  at  th(3  right  hand  or  place  of  units. 


liqUAilE    ROOT.  107 

2.  Begin  at  the  left  hand,  and  find  the  quotient  root  in  thai 
.  jriod,  and  place  it  on  the  right  of  the  given  sum  in  the  quo- 
wont,  and  its  square  under  said  period,  which  subtract  from  the 
number  above. 

3.  Then  bring  down  the  next  period  of  2  figures,  and  place 
it  on  the  right  of  the  remainder,  as  in  division,  and  this  forms 
a  new  dividend. 

4.  Now  double  this  figure  or  root  in  the  quotient,  and  place 
it  on  the  left  of  the  new  dividend  for  a  divisor. 

5.  Then  consider  how  often  the  divisor  is  contained  in  the 
dividend,  omitting  the  last  figure,  and  place  the  result  on  the 
right  of  the  root  in  the  quotient,  and  then  place  this  figure  on 
the  right  of  the  number  produced  by  doubling  for  a  divisor,  and 
multiply  as  in  division,  until  the  periods  are  all  brought  down. 

For  Decimals. — When  decimals  occur  in  the  given  number, 
it  must  be  pointed  both  ways  from  the  decimal  point,  and  the 
root  must  consist  of  as  many  figures,  of  whole  numbers  and 
decimals  respectively,  as  there  are  periods  of  integers  or  deci- 
mals in  the  given  number.  When  a  decimal  alone  is  given, 
annex  one  cipher,  if  necessary,  so  that  the  number  of  decimal 
places  shall  be  equal ;  and  the  number  of  decimal  places  in  the 
root  will  be  equal  to  the  number  of  periods  in  the  given  decimal 

For   Vulgar  Fractions. — 1.  Reduce   mixed  numbers  to  im 
proper  fractions,  and  compound  fractions  to   simple  ones,  and 
then  reduce  the  fraction  to  its  lowest  terms. 

2.  Extract  the  square  root  of  the  numerator  and  denominator 
separately,  if  they  have  exact  roots  ;  but  if  they  have  not,  reduce 
the  fraction  to  a  decimal,  and  then  extract  the  root,  as  above,  &c. 

Proof:  square  the  root  and  add  in  the  remainder. 

EXAMPLES. 

Illustration. — A  square  number  can  not  have  more  places  of 
figures  than  double  the  places  of  the  root,  and  at  least  but  one 
less.  A  square  is  a  figure  of  four  equal  sides,  each  pair  meet- 
ing perpendicularly,  or  a  figure  whose  length  and  breadth  are 
equal.  As  the  area,  or  number  of  equal  feet,  inches,  &c.,  in  a 
square,  is  equal  to  the  products  of  the  two  sides,  which  are 
equal,  the  second  power  is  called  the  square.  Let  the  follow^ 
ing  figure  represent  a  board  one  foot  square,  and  one  inch  in 
thickness,  which  being  sawn  or  cut  into  square  or  solid  inches, 
will  make  144  inches,  or  144  blocks  one  inch  square  ;  and  the 
square  root  of  144  is  12  ;  because  12  x  12  =  144,  which  in  this 
case  will  be  inches = one  side  of  the  square. 

17* 


198 


SQUARE    ROOT. 


Thus,  12  inches.  .     operation. 

-  raiSfflllBHBBHHHB  Z 

X  mnmmmmmmmmjam  ^ 

\  ^BBBHHHHBHB   ii 

\  IBBriBBIBHBHHHB  ^ 

.BBBBBBBBBBBH  "^ 

\  BillBBJBiiaBBBBil  g* 

^  HiJEBBeiBHBBBB  o 

aflflBBBBBBBBB  ^ 

I^LimSSSlBBBBflBB  % 

Figure  1. 
1.  Required  the  square  root  of  154682  ? 


144 
1 

22)44(12=1  foot,  or  length 
of  one  side  of  the  square  =:  12 
inches.  12x12  =  144  inches 
1  square  fool.  Proof 


Thus,     154682(393  root. 
3x3=      9 

3932 
393 

Explanation. — 
First    seek    the 

3X2  =  69)646 
621 

1179 
3537 

square  of  the 
first  period  (15), 
3X3  =  9,  which 

1179 

is  9,  it  can  not 

39x2=783)2582 
2349 

be  4,  because 
4x4  =  16  which 

154449 

233  rem. 

is  more  than  15  ; 

233  rem. 

place  the  3  in 
the  quotient  and 

154682  proof. 

the  9  under  the  first  period  (15),  which  subtract  and  you  have  6  ; 
then  bring  down  the  next  period  (46),  which  place  to  the  right 
of  the  remainder  (6)  ;  now  double  the  quotient  figure,  3x2  =  6, 
and  place  it  in  the  divisor,  one  place  to  the  left ;  now  consider 
how  many  times  6  in  64,  which  for  trial  call  9  ;  write  9  in  the 
quotient,  and  to  the  right  of  6  in  the  divisor,  which  makes  the 
divisor  69  ;  then  multiply  69  by  9,  and  place  the  result  under 
the  dividend  ;  subtract  as  before,  and  bring  down  the  next  pe- 
riod (82);  now  double  the  two  quotient  figures,  39  +  2  =  78, 
which  place  in  the  divisor ;  consider  how  many  times  78  is 
contained  in  258,  suppose  3  times  ;  place  3  in  the  quotient  and 
in  the  divisor,  as  before  ;  then  multiply  the  divisor  by  the  last 
quotient  figure,  and  write  the  result  as  above  directed,  and  the  re- 
mainder is  233  ;  which  by  annexing  ciphers  may  be  continued. 

2.  What  is  the  square  root  of  74770609  ?  Arts.  8647 

3.  What  is  the  square  root  of  54990.25  ?  234.5, 

4.  What  is  the  square  root  of  .3271.4007?  57.19-h 

5.  What  is  the  square  root  of  14876.2357?  121.968175. 

6.  What  is  the  square  root  of  96385103  ?  9817  + 


SQUARE  ROOT.  199 

7.  What  is  the  square  root  of  4.372594  ?  Ans.  2.091  + 

8.  What  is  the  square  root  of  .00103041  ?  Ans,  .0321. 

9.  What  is  the  square  root  of  f|§i?  Ans.  |. 

10.  What  is  the  square  root  of  IJf  1  A7is.  .89802+ 

1 1 .  What  is  the  square  root  of  -}  ?  A7is.  -|. 
^2.  What  is  the  square  root  of  6^  ?                                 Ans.  2^. 

13.  What  is  the  difference  between  ^81  and  Sl^? 

Ans.  6552. 

14.  What  is  the  square  root  of  |-|-^?  Ans.  |. 

15.  What  is  the  square  root  of  J-^-?  Ans.  .95744- 

16.  What  is  the  square  root  of  912^^5-?  Ans.  30-}. 

17.  What  is  the  square  root  of  2-1?  Ans.  1.5. 

18.  What  is  the  square  root  of  9980.01  ?  Ans.  99.9. 

19.  What  is  the  square  root  of  2  ?  Ans.  1.414213. 

20.  What  is  the  square  root  of  ^J{|^  ?        Ans.  ^^^  =  .09756  + 

21.  What  is  the  square  root  of  .0000316969?       Ans.  .00563. 

22.  What  is  the  square  root  of  964.5192360241  ? 

Ans.  31.05671. 

23.  What  is  the  square  root  of  m|  ?  Ans.  yV- 

24.  What  is  the  square  root  of  ^^-|^  ? 

25.  What  is  the  square  root  of  6.9169  ?  Ans.  2.63. 

26.  What  is  the  square  root  of  106929  ?  Ans.  327. 

27.  What  is  the  square  root  of  -if  ?  Ans.  ^. 

28.  What  is  the  square  root  of  ^l  Ans.  .7745. 

29.  What  is  the  square  root  of  20A  ?  Ans.  4-^. 

APPLICATION. 

To  find  a  mean  proportional  between  two  numbers. 

RULE. 

Multiply  the  given  numbers  together,  and  extract  the  square 
root  of  the  product,  which  will  be  the  mean  proportional  sought. 

30.  What  is  the  mean  proportional  between  24  and  96  ? 

Ans.  ^96X24=48. 

To  find  the  side  of  a  square  equal  in  area  to  any  given  superficies 
whatever. 

RULE. 

Find  the  area,  and  the  square  root  is  the  side  of  the  square 
sought. 

31.  If  the  area  of  a  circle  be  184.125,  what  is  the  side  of  a 
square  equal  in  area  thereto  ?  Ans.  y^l84. 125  =  13.569+ 


200 


SQUARE    ROOT. 


32.  A  general  has  an  army  of  5625  men  ;  how  many  must  h« 
place  in  ranlc  and  file  to  form  them  into  a  square  ? 

Ans.  ^5625  —  75. 

33.  Suppose  that  Napoleon,  at  the  battle  of  Marengo,  com- 
manded an  army  of  256036  men ;  how  many  did  he  place  in 
rank  and  file  to  form  them  into  a  solid  square  ?  Ans.  506. 

Having  the  area  of  a  circle  to  find  the  diameter. 


Multiply  the  square  root  of  the  area  by  1.12837,  and  the 
product  will  be  the  diameter.  Or  multiply  the  area  by  1 .2732 
and  take  the  square  root  of  the  product. 

34.  Required  the  diameter  of  a  circle  whose  area  is  82  feet 
81  inches.  Ans.  10  feet  3.13  inches. 

35.  Admit  a  leaden  pipe  |  of  an  inch  in  diameter  will  fill  a 
cistern  in  3  hours  ;  I  demand  the  diameter  of  another  pipe  which 
will  fill  the  same  cistern  in  1  hour. 

Thus  f=.75  and  .75  X  .75^.5625 ;  then  3  hours  :  5625  ::  1 
hour:  1.6875  and ^1.6875=1.3  inches  nearly,  ^n^.  (inversely.) 

36.  If  a  circular  pipe  of  1.5  inches  diameter  fill  a  cistern  in 
5  hours,  in  what  time  would  it  be  filled  by  one  3.5  inches  di- 
ameter? Ans.  55  minutes,  4.8  seconds. 

37.  If  784  trees  be  planted  in  a  square  orchard,  how  many 
must  be  in  a  row  ?  Ans.  28. 

The  square  of  the  longest  side  or  hypotenuse  of  a  right-angled 
triangle,  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides ; 
consequently,  the  difference  of  the  square  of  the  longest,  and  either 
of  the  others,  is  the  square  of  the  remaining  side. 

38.  The  wall  of  a  fort  is  17  feet  high,  which  is  surrounded 
by  a  ditch  20  feet  in  breadth  ;  required  the  length  of  a  ladder  to 
reach  from  the  outside  of  the  ditch  to  the  top  of  the  wall. 

^n^.  17xl7=i289  :  20x20=400+ 289  =  ^689=26.2-1- 


^^. 


ditch  20  feet. 


SQUARE     ROOT. 


201 


39.  Two  ships  leave  tlie  same  port,  one  sails  due  east  40 
miles'  and  the  other  due  north  50  miles  ;  required  the  distance 
irom  each  other. 


40  m.  East. 


40.  There  is  a  church  48  feet  in  height  from  the  ground  to 
the  eaves  or  roof,  and  the  street  is  64  feet  in  width  ;  what  length 
of  cord  would  it  require  to  reach  from  the  opposite  side  of  the 
street  to  the  eaves  of  the  house  ?  Ans.  80  feet. 

41.  In  a  circle,  whose  area  or  superficial  content  is  4096 
feet,  I  demand  what  will  be  the  length  of  one  side  of  the  square  . 
containing  the  same  number  of  feet.  Ans.  64  feet. 

42.  A  certain  square  garden  measures  4  rods  on  each  side  ; 
what  will  be  the  length  of  one  side  of  a  garden  containing  4 
times  as  many  square  rods  ?  Ans.  8  rods. 

43.  If  one  side  of  a  square  piece  of  land  measure  5  rods, 
what  will  be  the  side  of  one  measuring  4  times  as  large  ?  16 
times  as  large  ?   36  times  as  large  ?  Ans.  10,  20,  30. 

44.  In  a  load  of  120  cherry-boards  of  one  inch  in  thickness, 
27  inches  in  width,  and  11.5  feet  in  length,  how  many  square 
feet,  and  what  is  the  value  of  the  boards  at  D22  per  M.  ? 

Ans.  3105  feet;  value  D68.31. 

45.  In  a  tract  of  land  of  640  acres,  required  the  length  of  one 
side  ;  and  suppose  the  tract  to  contain  16  times  as  much,  what 
would  be  the  length  of  one  side  ?  and  suppose  each  rod  valued 
at  D75,  what  would  be  the  amount  ? 

Ans.  1st,  320  poles,  length  of  one  side  ;  2d,  1280  po.,  or  16 
times  as  much  ;  3d,  D 122880000  value. 

46.  One  side  of  a  square  field  is  400  rods  in  length  ;  required 
the  number  of  acres  in  the  field. 

Note. — For  examples  in  square  and  cub.  measure,  see  appendix. 

REVIEW. 

What  is  evolution  ?  What  is  the  square  root  of  a  number  ? 
What  is  the  cube  root  ?  When  you  wish  to  extract  the  square 
root  of  a  number,  what  must  first  be  done  ?  After  you  have 
counted  otT  the  given  number  into  periods,  how  will  you  pro- 


202 


SQUARE    ROOT. 


ceed  ?  Repeat  the  rule.  When  the  given  number  is  a  whole 
number,  how  many  figures  will  there  be  in  the  root?  When 
you  can  not  get  the  exact  root,  what  can  be  done  ?  How  can 
you  extract  the  root  of  a  decimal  fraction  1  What  will  you  do 
with  a  whole  number  and  decimal  1  What  is  the  rule  ?  How 
will  5^ou  extract  the  square  root  of  a  vulgar  fraction  ?  Repeat 
the  rule  ?  Give  an  example  on  the  slate.  What  is  the  differ- 
ence between  a  square  and  the  square  root  ? 

47.  Required  the  square  root  of  9876.4792 iy*2*  ?     (This  is 
given  for  a  trial  question.) 


CUBE  ROOT. 

The  cube  of  a  number  is  the  product  of  that  number  multiplied 
into  its  square.  To  extract  the  cube  root,  is  to  find  a  number 
which  being  multiplied  into  itself,  and  then  into  that  product, 
will  produce  the  given  number. 

ILLUSTRATION. 

The  solid  called  a  cube,  has  its  length,  breadth,  and  thick- 
ness, all  equal.  As  the  number  of  solid  feet,  inches,  &c.,  in  a 
cube,  are  found  by  multiplying  the  length,  height,  and  breadth, 
together — that  is,  by  multiplying  one  side  into  itself  twice,  the 
third  power  of  a  number  is  called  the  cube  of  that  number  ;  thus 
a  cubical  or  solid  foot  has  6  equal  sides  of  1  foot  square,  con- 
sequently it  follows  that  a  cubical  foot  contains  1728  solid  or 
cubical  inches,  because  12  X  12=144  X  12=1728  inches  :  that 
is,  a  cubical  foot  or  block  of  wood  may  be  sawn  into  1728 
blocks,  each  of  which  will  be  a  solid  inch,  which  may  be  rep- 
resented in  the  following  manner  : — 

2  3  4  5  6 


144  144 


144X12=^ 


1  solid  foot. 


u:jji:i::::    'iJiiiiJlUS    sHiJlKf^   rI!I:ku!I  :::~::n:i:'    i::::;::;::; 


12        6  equal  sides 


144 


144 


CUBK    ROOT.  203 

Figure  2  is  a  solid  or  cubical  foot  with  6  equal  sides  ;  sup- 
pose this  block  be  sawn  into  12  pieces  at  the  distance  of  1  inch, 
it  would  make  12  pieces,  as  represented  above,  each  1  foot 
square,  and  1  inch  in  thickness,  as  in  figure  1  (S.  R.)  ;  now, 
each  of  those  12  square  pieces  may  be  divided  into  144  solid 
blocks:  the  12  blocks  divided  would  be  12x12  =  144  inches 
for  each  of  the  12  pieces,  as  represented  by  tbe  12  squares, 
which  is  equal  to  1728  inches  in  a  cubical  or  solid  foot,  and  12 
is  the  cube  root  of  1728. 

Thus,  the  root  12  is  equal  to  12  square  feet  of  ]  inch  in  thick- 
ness (as  1,  2,  3,  &c.),  or  12  inches  in  length,  breadth,  and 
thickness,  which  is  equal  to  1  solid  foot. 

Thus,  1x300=300  :  2x30=60  :  2x2=4 
60 
4+  1728(12  root. 

364  divisor,  

364)  728 

728 

In  the  example  above,  I  first  point  off  the  three  figures  from 
the  right,  and  place  the  period  over  the  7  ;  this  leaves  1,  or  unity, 
for  the  first  operation  ;  consequently  the  quotient  figure  must  in 
this  case  be  1,  which  I  put  in  the  quotient,  and  under  1  in  the 
dividend  ;  then  bring  down  the  next  period  (728) ;  now  seek  for 
the  divisor  ;  first  multiply  1  by  300  =  300  for  a  trial  divisor,  then 
suppose  300  in  728,  say  2,  which  put  in  the  quotient;  then 
2  X  30=60  ;  then  2x2  =  4  the  square  of  2  ;  then  add  these  sev- 
eral products  together  and  you  have  the  true  divisor  (364)  which 
multiply  by  the  last  quotient  figure  (2)  and  the  work  is  finished. 


1 .  Point  the  given  number  into  periods  of  three  figures  each, 
beginning  at  the  right  or  units'  place. 

2.  Find  the  greatest  cube  in  the  left-hand  period,  and  sub- 
tract it  therefrom,  and  set  do\vn  the  root  in  the  quotient,  and 
then  to  this  remainder  bring  down  and  annex  the  next  period 
for  a  dividend. 

3.  Square  the  quotient  by  multiplying  it  into  itself,  and  mul- 
tiply that  product  by  300  for  a  trial  divisor ;  see  how  often  it  i? 
contained  in  the  dividend,  and  set  the  result  in  the  quotient. 

4.  Multiply  the  figure  last  put  in  the  quotient  by  the  other 
figures  in  the  quotient,  and  that  product  by  30. 


204  CUBE    ROOT. 

5.  Then  square  this  last  figure  in  the  quotient,  and  add  it  to 
ihe  product  just  mentioned,  for  the  second  part  of  the  divisor; 
all  these  products  added,  form  the  true  divisor. 

6.  Now  multiply  this  true  divisor  by  the  last  figure  put  in  the 
quotient ;  subtract  the  product  from  the  dividend  as  in  division 
bring  down  the  next  period  of  three  figures  ;  proceed  as  before. 

For  Decimals, — Annex  ciphers  to  the  decimal,  if  necessary, 
so  that  it  shall  consist  of  3,  6,  9,  &c.,  places ;  then  put  the  first 
point  over  the  place  of  thousandths,  the  second  over  the  place 
of  millionths,  and  so  on  over  every  third  place  to  the  right ;  and 
then  extract  the  root  as  in  whole  numbers.  Observe,  1.  There 
will  be  as  many  places  in  the  root  as  there  are  periods  in  the 
given  number.  2.  The  same  rule  applies  when  the  given  num- 
ber is  composed  of  a  whole  number  and  a  decimal.  3.  If  there 
be  a  remainder  in  a  whole  number,  after  all  the  periods  have 
been  brought  down,  you  can  annex  periods  of  ciphers,  consid- 
ering them  as  decimals. 

For  Vulgar  Fractions. —  1.  Reduce  compound  fractions  to 
simple  ones,  mixed  numbers  to  improper  fractions,  and  then  re- 
duce the  fraction  to  its  lowest  terms.  2.  Then  extract  the  cube 
root  of  the  numerator  and  denominator  separately,  if  they  have 
exact  roots  ;  but  if  either  of  them  have  not  an  exact  root,  reduce 
the  fraction  to  a  decimal,  and  extract  the  root  as  above. 

Proof. — Cube  the  root  and  add  in  the  remainder. 

EXAMPLES. 

1.  What  is  the  cube  root  of  5382674  ?     .      . 

5382674(175.2  root, 
i  \^Ans 


1X1X300=300 
1X7X    30=210  + 
7x7            =49 

17X17X300  = 

17x    5X    30  = 

5x    5            = 

86700      ) 
2550+  \ 
25      )     = 

=559)4382 
3913 


:89275)469674 
446375 


1752x300  =  9187500 
175x2x30=10500  + 
2X2  =  4 


od  Divisor,         9198004, 


9198004)23299000 
18396008 


4902992  rem. 


RULE    II. 

1.  Point  off  the  given  number  into  periods  of  three  figures 
each,  if  whole  numbers  commencing  at  the  right  haiid ;  but  if 
decimals,  at  the  left. 


CUBE    ROOT. 


2Qd 


2.  Find  the  greatest  cube  in  the  left  hand  period,  and  place 
ihe  root  to  the  right  of  the  given  number,  and  subtract  the  cube 
of  the  root  from  the  left  hand  period,  and  to  the  remainder  bring 
down  the  next  period  for  a  dividend. 

3.  Square  the  root,  and  multiply  it  by  three  for  a  defective 
divisor. 

4.  Reserve  mentally  the  units  and  tens  of  the  dividend,  and 
try  how  often  the  defective  divisor  is  contained  in  the  remainder; 
place  this  result  to  the  root,  and  the  square  of  it  to  the  right  of 
the  defective  divisor,  but  if  the  square  is  less  than  ten,  supply 
the  ten's  place  by  a  cipher. 

5.  Complete  the  divisor  by  adding  thereto  the  product  of  the 
last  figure  of  the  root,  and  the  remaining  figure  or  figures  of  the 
root,  and  that  again  by  30  ;  then  divide  and  subtract  as  in  long 
division. 

6.  The  defective  divisors  after  the  first  may  be  easily  found 
by  adding  to  the  last  divisor  the  number  that  completed  it,  with 
twice  the  square  of  the  last  figure  of  the  root. 

7.  The  divisor  is  then  completed  according  to  the  4th  and 
5th  paragraphs  above. 

2.   V3796416  .      . 


Greatest  cube  of  3  is  1. 
Square  of  1  multiplied  by 
the  square  of  5  added,  is 
The  5  paragraph  produces 

3  and 

325 

=  150 

3796416( 

1 

2796 
2375 

156 

Complete  divisor 
The  number  that  com- 

=475 

421416 

pleted  it. 

150 

421416 

Twice  the  square  of  the 
last  figure  of  the  root. 

50 

Square  of  68,  defective 
divisor. 

67536 

Paragraph  5  produces, 

2700 

Complete  divisor, 

70236 

3.  What  is  the  cube  root 

of  99252.847 

? 

4.  What  is  the  cube  root 

of  259694072 

? 

Ans.  638. 

5.  What  is  the  cube  root  of  34328125  ? 

325. 

6.  What  is  the  cube  root 

of  12.977875 

? 

2.35. 

7.  What  is  the  cube  root 

of  171.46776406? 

8    What  is  the  cube  root  of  .5  ? 

5.555  + 

17+87  rem 

18 

206  CUBE    ROOT. 

9.  What  is  the  cube  root  of  32461759  ?  Ans.  319, 

10.  What  is  the  cube  root  of  27054036008  ?  3002 

11.  What  is  the  cube  root  of  .751089429?  .0909. 

12.  What  is  the  cube  root  of  3.408862625  1  1.505 

13.  What  is  the  cube  root  of  |ff  ?  ^,  \, 

14.  What  is  the  cube  root  of  31  J/g  ?  3f  "' 

15.  What  is  the  cube  root  of  f  ?  .822-f 

16.  What  is  the  cube  root  of  |fif  1                     '  ^^ 

17.  What  is  the  cube  root  of  -j^q  ?  13  + 

18.  What  is  the  cube  root  of  84.604519?  4.39, 

19.  What  is  the  cube  root  of  ^-||  ?  f 

20.  What  is  the  cube  root  of  16194277?  253. 

21.  What  is  the  cube  root  of  54854153  ?  379.958793  + 

22.  What  is  the  cube  root  of  i||M 

23.  What  is  the  cube  root  of  j\^^j  1  y\. 

24.  What  is  the  cube  root  of  7  1  1.9129. 

25.  What  is  the  cube  root  of  15625  ?  25 

26.  What  is  the  cube  root  of  436036424287  ? 

27.  What  is  the  cube  root  of  99  ?  4.62+ 


APPLICATION. 

To  find  two  mean  proportionals  between  any  two  given  numbers 

RULE. 

Divide  the  greater  by  the  less,  and  extract  the  cube  root  of 
the  quotient. 

2.  Multiply  the  root  so  found  by  the  least  of  the  given  num- 
bers, and  the  product  will  be  the  least. 

3.  Multiply  this  product  by  the  same  root,  and  it  will  give  the 
greatest. 

28.  What  are  the  two  mean  proportionals  between  6  and 
750? 

Thus:  750-4-6=125  and  3/125  =  5,  then  5x6  =  30,  least,* 
and  30x5  =  150  greatest;  30  and  150.  Ans, 

Note. — The  solid  contents  of  similar  figures  are  in  proportion 
to  each  other,  as  the  cubes  of  their  similar  sides  or  diameters. 

29.  If  a  bullet  6  inches  in  diameter  weigh  32  lb.,  what  will 
a  bullet  of  the  same  metal  weigh,  whose  diameter  is  3  inches  ? 

As6x6x6=216;  3x3x3=27;  as216:  32 lb.  ::  27  :  4  lb. 


CUBE    ROOT.  207 

The  side  of  a  cube  being  given,  to  find  the  side  of  that  cube  which 
shall  be  double ,  triple,  <^c.,  in  quantity  to  the  given  cube. 

RULE. 

Cube  your  given  side,  and  multiply  it  by  the  given  propcrtion 
between  the  given  and  required  cube,  and  the  cube  root  of  the 
product  will  be  the  side  sought. 

30.  If  a  cube  of  silver  whose  side  is  4  inches,  be  worth  D50, 
I  demand  the  side  of  a  cube  of  the  like  silver,  whose  value 
shall  be  4  times  as  much  ? 

^725.  4X4X4  =  64,  and  64x4=256  :   V256=6-349  +  in.ches. 

31.  There  is  an  oblong  cellar,  the  content  of  which  is 
1953.125  cubic  feet ;  what  is  the  side  of  a  cubical  cellar  that 
shall  contain  just  as  much?  Ans.  12.5  feet. 

32.  What  is  the  difTerence  between  a  solid  half  foot  and  half 
of  a  solid  foot  1  Ans.  3  half  feet. 

33.  In  221184  solid  inches  how  many  cords  1     Ans.  1  cord. 

34.  Admitting  a  room  to  be  11  feet  high,  21  feet  in  length, 
and  16  feet  in  width,  what  number  of  cubic  feet  of  space  in  it ' 

^^_  Ans.  3696  cubic  feet. 

|^K35.  The  diameter  of  a  bushel  measure  being  18^  inches,  and 

^flre  height  8  inches,  what  is  the  side  of  a  cubic  box  which  shall 

contain  that  quantity  ?  Ans.  12.907+inches. 

36.  In  a  cubic  foot,  how  many  cubes  of  6  inches,  and  how 
many  of  4,  of  3,  of  2,  of  1,  are  contained  therein  ? 

Ans.  8  of  6  inches*;  27  of  4  inches  ;  64  of  3  inches  ;  216 
of  2  inches  ;  1728  of  1  inch. 

37.  Suppose  a  cubical  cellar  to  contain  1728  solid  feet,  what 
will  one  of  its  cubic  sides  measure  ? 

38.  In  a  square  box  that  will  contain  1000  marbles,  how  many 
will  it  take  to  reach  across  the  bottom  of  the  box,  in  a  straight 
row  ?  Ans.  10. 

39.  What  is  the  difference  between  the  cube  root  of  27  and 
the  square  root  of  9  ?  Ans.  0 

40.  If  a  globe  of  silver  3  inches  in  diameter  be  worth  D160, 
what  is  the  value  of  one  6  inches  in  diameter  ? 

Ans.  3^   :  6^  ::  D160  :  D1280. 

41.  If  the  diameter  of  the  planet  Jupiter  is  12  times  as  much 
as  the  earth,  how  many  globes  of  the  earth  would  it  take  to 
make  one  as  large  as  Jupiter  ?  Ans.  1728. 

42.  There  are  two  small  globes  ;  one  of  them  is  one  inch  in 
diameter,  and  the  other  two  inches ;  how  many  of  the  smaller 
globes  will  make  one  of  tbe  larger  ?  Ans.  8. 


208  ALLIGATION   MEDIAL. 


RULE    III. 


1.  Find  the  quotient  root  of  the  left  hand  period,  which  sub- 
tract  from  the  same,  and  then  bring  down  the  next  period.  2. 
Multiply  the  square  of  the  quotient  figure  by  300  for  a  divisor ; 
then  find  the  next  figure ;  square  this  quotient  figure  ;  multiply 
that  square  by  the  other  quotient  figure,  and  then  by  30  ;  find 
the  cube  of  this  last  quotient  figure  ;  add  both  these  products  to 
the  product  of  the  divisor  and  the  quotient  figure,  the  sum  of 
which  subtract  from  the  dividend.  3.  Then  bring  down  the 
next  period,  which  will  complete  the  next  dividend  ;  square 
your  two  quotient  figures,  and  multiply  by  300  for  your  next 
divisor,  and  so  continue  till  the  operation  is  completed. 

Note. — The  roots  of  the  4,  6,  8,  9,  and  12  powers  may  be 
obtained  in  the  following  manner  : — 

For  the  4th  root,  take  the  square  root  of  the  square  root. 

For  the  6th,  take  the  square  root  of  the  cube  root. 

For  the  8th,  take  the  square  root  of  the  4th  root. 

For  the  9th,  take  the  cube  root  of  the  cube  root. 

For  the  12th,  take  the  cube  root  of  the  4th  root,  &c. 


What  is  a  cube  ?  What  is  the  cube  root  7  What  will  you  first 
do  to  extract  the  cube  root  of  a  whole  number  ?  Repeat  the  rule  ? 
How  do  you  extract  the  cube  root  of  a  decimal  fraction  ?  When 
there  is  a  decimal  and  whole  number,  how  will  you  point  them 
off?  Why  do  you  point  decimals  from  the  left  or  decimal  point 
toward  the  right  ?  In  extracting  a  root,  if  there  be  a  remainder, 
what  may  be  done  ?  What  is  the  rule  for  decimals  ?  How  do 
you  extract  the  cube  root  of  a  vulgar  fraction  ?  What  is  the 
rule  ?  How  do  you  extract  the  cube  root  of  a  mixed  number  ? 
What  is  the  difference  between  a  cube  and  the  cube  root  ? 

43.  What  is  the  cube  root  of  36|f  1  Ans.  3.32  + 


ALLIGATION  MEDIAL. 

Alligation  is  used  when  the  quantities  and  prices  of  several 
things  are  given,  to  find  the  mean  price  of  the  mixture  composed 
of  those  materials  ;  or  the  method  of  mixing  two  or  more  sim- 
ples of  different  qualities,  so  that  the  composition  may  be  of  a 
mean  or  middle  quality.  There  are  two  kinds,  Alligation  Me- 
dial and  Alligation  Alternate. 


ALLIGATION    ALTERNATE.  209 


Divide  the  entire  cost  of  the  whole  mixture  by  the  sum  of  the 
simples  ;  the  quotient  will  be  the  price  of  the  given  mixture. 

1.  If  19  bushels  of  wheat  at  75c.  per  bushel,  40  bushels  of 
rye  at  50c.  per  bushel,  and  12  bushels  of  barley  at  37.5c.  per 
bushel,  be  mixed  together,  what  is  a  bushel  of  the  mixture 
worth?  19x75   =14.250. 

40x50  =20.00 
12X37^=  4.50 


71  71)38.75(54c.  6m.  Ans. 

2.  If  4  ounces  of  silver,  worth  62.5c.  per  ounce,  be  melted 
with  8  ounces  at  50c.  per  ounce,  what  is  an  ounce  of  this  mix- 
ture worth?  Ans.  54c. 

3.  A  goldsmith  melted  together  8  ounces  of  gold,  22  carats 
fine  ;  1  lb.  8  oz.  of  21  carats  fine  ;  and  10  oz.  of  1 8  carats  fine  ; 
what  is  the  quality  of  fineness  of  this  composition  ? 

Ans.  20y^^  carats  fine. 

4.  In  buying  tea,  I  pay  90c.  per  lb.  for  12  lbs.  and  Dl.20  per 
lb.  for  16  lbs.;  what  is  the  mixture  worth  per  lb.  ?  Ans.  Dl  .07.1  -|- 

5.  A  wine  merchant  mixed  12  gallons  of  wine,  at  75c.  per 
gallon,  with  24  gallons  at  90c.  per  gallon,  and  16  gallons  at 
Dl.lO;  what  is  1  gallon  worth?  Ans.  92c.  6m. 

6.  On  a  certain  day  the  mercury  in  the  thermometer  was  ob- 
served to  average  the  following  heights  :  from  6  in  the  morning 
to  9,  64°  ;  from  9  to  12, 74^  ;  from  12  to  3,  84^  ;  and  from  3  to  6, 
70^  ;  what  was  the  mean  temperature  of  the  day  ?       Ans,  73^. 


ALLIGATION  ALTERNATE 

Is  the  method  of  finding  what  quantity  of  each  of  the  ingre- 
dients whose  rates  are  given,  will  compose  a  mixture  of  a  given 
rate  ;  so  that  it  is  the  reverse  of  Alligation  Medial,  and  may  be 
proved  by  it. 

To  find  the  proportion  in  vMch  several  simples  of  given  prices 
must  be  mixed  together,  that  the  compound  may  he  worth  a  given 
price. 


W  1.  Set  down  the  prices  of  the  simples  under  each  other  iu 
the  order  of  theii  values,  beginning  with  the  lowest 

1^* 


210 


ALLIGATION*    ALTERNATE. 


2.  Link  the  least  price  with  the  greatest,  and  the  next  least 
with  the  next  greatest,  and  so  on,  until  the  price  of  each  simple 
which  is  less  than  the  price  of  the  mixtures  is  linked  with  one 
or  more  that  is  greater,  and  every  one  that  is  greater  with  one 
or  more  that  is  less. 

3.  Write  the  difference  between  the  price  of  the  mixture  and 
that  of  each  of  the  simples  opposite  that  price  with  which  the  par- 
ticular simple  is  linked  ;  then  the  difference  standing  opposite 
any  one  price,  or  the  sum  of  the  differences  when  there  is  more 
than  one,  will  express  the  quantity  to  be  taken  at  that  price. 

1.  If  you  would  mix  wines  worth  16c.,  18c.,  and  22c.  per 
quart  in  such  a  way  that  the  mixtures  be  worth  20c.  quart,  how 
much  must  be  taken  of  each  sort  1 

Thus:  (^^1  2  at  16c.  )2  qts.  at  16c.  )    . 

given  price  20c.  <  18s  2  at  18c.  V  2     "        18c.  >  § 

(  22'' -J  4  +  2  =  6  at  22c.  )  6     "        22c.  )  ^ 

2.  How  much  corn  at  42c.,  60c.,  67c.,  and  78c.,  per  bushel, 
must  be  mixed  together  that  the  compound  may  be  worth  64c. 
per  bushel  ? 

Ans,  14  bushels  at  42c.,  3  bushels  at  60c.,  4  bushels  at  67c. ^ 
and  22  bushels  at  78c. 

3.  What  quantity  of  oats  at  50c.  per  bushel,  and  30c.  per 
bushel,  must  be  mixed  together,  that  the  compound  may  be 
worth  40c.  per  bushel  ?        Ans.  an  equal  quantity  of  each  sort. 

4.  How  much  rye  at  50c.  per  bushel,  barley  at  37^c.  per 
bushel,  and  oats  at  25c.  per  bushel,  will  make  a  mixture  worth 
31c.  per  bushel? 

Ans.  6  bushels  at  50c.,  6^  at  37ic.,  aud  25^  at  25c. 
JS^ote. — If  all  the  given  prices  are  greater  or  less  than  the  mean 
or  given  price,  they  must  be  linked  to  a  cipher.     A  variety  of 
answers  may  be  obtained,  according  to  the  method  of  linking. 

RULE    II. 

1.  Find  the  proportional  quantities  of  the  simples  as  in  rule 
1st.  2.  Then  say  as  the  number  opposite  the  simples  whose 
quantity  is  given,  is  to  the  given  quantity,  so  is  either  propor- 
tional quantity  to  the  part  of  its  simple  to  be  taken. 

5.  What  quantity  of  coffee  at  20c.  and  at  16c.  must  be  mixed 
with  35  lbs.  at  14c.  per  lb.  to  make  a  mixture  worth  18c.  per  lb.  ? 

(  14  V  2—  :  35  ::  2  :     35lbs.  at  16c.  ) 

18  }  16.  2—  :  35  ::  6  :  lOSlbs.  at  20c.  V 

^20''>'  4+2=6  ) 

For  35x2-2=  35  at  16c.  )   . 

and  35x6-2  =  105  at  20c.  J  ^^* 


ALLLIGATION    ALTERNATE.  311 

6.  A  farmer  would  mix  14  bushels  of  wheat  at  D1.20  per 
bushel,  with  rye  at  72c.,  barley  at  48c.,  and  oats  at  36c. ;  how 
much  must  be  taken  of  each  sort  to  make  the  mixture  worth 
64c.  per  bushel  ? 

Ans.  14  bushels  of  wheat,  8  bushels  of  rye,  4  bushels  of  bar- 
ley, and  28  bushels  of  oats. 

7.  A  person  wishes  to  mix  10  bushels  of  wheat  at  70c.  per 
bushel,  with  rye  at  48c.,  corn  at  36c.,  and  barley  at  30c.  per 
bushel,  so  that  a  bushel  of  this  mixture  may  be  worth  38c.  ; 
what  quantity  of  each  must  be  taken  ? 

Ans.  2J  bushels  of  rye,  12J  bushels  of  corn  ;  40  bushels  of 
barley. 

8.  How  much  water  must  be  mixed  with  100  gallons  of  wine 
worth  90c.  per  gallon,  to  reduce  it  to  75c.  per  gallon. 

Ans.  20  gallons. 

When  the  quantity  of  the  compound  is  given  as  well  as  the  price* 

RULE    III. 

1.  Find  the  proportional  quantities  as  in  rule  1st. 

2.  Then  say,  as  the  sum  of  the  proportional  quantities  is  to 
the  given  quantity,  so  is  each  proportional  quantity  to  the  part 
to  be  taken  of  each. 

9.  A  grocer  has  currants  at  4c.  6c.,  9c.,  and  lie,  per  pound, 
and  he  wishes  to  make  a  mixture  of  2401b.,  worth  8c.  per 
pound  ;  what  quantity  of  each  kind  must  be  taken  ? 

'4—-.  —3  " 


8 


I] 

Lii- 


~2    Then 


10 

3  : 

:  240 

72lbs. 

at    4c. 

10 

1   : 

:  240 

241bs. 

at    6c. 

10 

2  ; 

:  240 

481bs. 

at    9c. 

10 

4  : 

:  240 

961bs. 

at  lie. 

^Ans. 


10 

10.  How  much  water,  at  0  per  gallon,  must  be  mixed  with 
wine  at  80c.  per  gallon,  so  as  to  fill  a  vessel  of  90  gallons, 
which  may  be  offered  at  50c.  per  gallon  ? 

Ans.  56^  gallons  of  wine,  33|  gallons  of  water. 

11 .  A  goldsmith  has  several  sorts  of  gold,  namely  :  of  15, 17, 
20,  and  22  carats  fine,  and  would  melt  together,  of  all  these 
sorts,  so  much  as  may  make  a  mass  of  40oz.,  18  carats  fine  ; 
how  much  of  each  sort  is  required  ? 

Ans.  16oz.  of  15  carats  fine,  8oz.  of  17,  4oz.  of  20,  12oz.  of  22. 

12.  How  much  barley  at  40c.  per  bushel,  rye  at  60c.,  and 
wheat  at  80c.,  must  be  mixed  together  that  the  compound  may  be 
worth  62^0.  per  bush.  ?    Ans.  17^  bu.  barley,  1 7^  rye,  25  wheat. 


212  ARITHMETICAL    PROGRESSIOX. 

13.  A  composition  was  made  of  5  lbs.  of  tea  at  Dl^  per  lb., 
9  lbs.  at  D1.80  per  lb.,  and  17  lb§.  at  Dl-^  per  lb.  ;  what  is  a 
pound  of  it  worth?  Ans.  Dl.54.6.f 

14.  A  grocer  would  mix  different  quantities  of  sugar,  name- 
ly :  one  at  20c.,  one  at  23c.,  and  one  at  26c. ;  what  quantity  ol 
each  sort  must  be  taken  to  make  a  mixture  worth  22c.  per  lb.  ? 

Ans.  5  lbs.  at  20c.,  2  at  23c.,  and  2  at  26c. 
Demonstration, — By  connecting  the  less  rate  with  the  greater, 
and  placing  the  difference  between  them  and  the  mean  rate  al- 
ternately, or  one  after  the  other,  in  turn  ;  the  quantities  are  such, 
that  there  is  precisely  as  much  gained  by  one  quantity  as  is  lost 
by  the  other,  and  therefore  the  gain  and  loss,  upon  the  whole, 
are  equal,  and  exactly  the  proposed  rate.  In  like  manner.  Jet 
the  number  of  simples  be  what  it  may,  and  with  how  many 
soever  each  one  is  linked,  since  it  is  always  a  less  with  a  greater 
than  the  mean  price,  there  will  be  an  equal  balance  of  loss  or 
gain  between  every  two,  and  consequently  an  equal  balance  on 
the  whole.     The  rule  is  founded  on  the  principles  of  proportion. 

REVIEW. 

What  is  Alligation  ?  What  is  Alligation  Medial  ?  How  do 
you  find  the  price  of  the  mixture  ?  What  is  the  rule  ?  What 
is  Alligation  Alternate  ?  How  can  you  prove  it  1  How  do  you 
find  the  proportional  parts  when  the  price  only  is  given  ?  Re- 
peat rule  1st.  What  is  the  rule  when  a  given  quantity  of  one 
of  the  simples  is  to  be  taken  ? 

What  is  the  rule  when  the  quantity  of  the  compound  as  well 
as  the  price  is  given  ?     What  more  can  you  say  of  Alligation  ? 

15.  Bought  a  pipe  of  wine,  containing  120  gallons,  at  Dl.30 
a  gallon ;  how  much  water  must  be  mixed  with  it  to  reduce  the 
first  price  to  Dl^O  a  gallon  ?  Ans.  21  j\  gallons. 

ARITHMETICAL  PROGRESSION. 

Any  series  of  numbers  more  than  two,  increasing  or  decreas- 
ing by  an  even  ratio  or  common  difference,  is  in  Arithmetical 
Progression.  When  the  numbers  are  formed  by  continual  ad- 
dition of  the  ratio  or  common  difference,  they  form  an  ascending 
series ;  but  when  formed  by  continual  subtraction  of  the  com- 
mon difference,  it  is  a  descending  series, 
rp,       (  0,  2,  4,  6,  8,  10,  &c.,  is  an  ascending  arithmetical  series. 

I  1,  2,  4,  8,  16,  32,        is  an  ascending  geometrical  series. 
.    T    C  10,  8,  6,  4,  2,  0,  is  a  descending  arithmetical  series. 

I  32,  16,  8,  4,  2,  1,        is  a  descending  geometrical  series. 


ARITHMETICAL    PROGRESSION.  213 

The  numbers  which  form  the  series  are  called  the  terms  of 
the  progression. 

The  first  and  last  terms  of  a  progression  are  called  the  ex- 
tremes, and  the  other  terms  the  means.  Any  three  of  the  fol 
lowing  things  being  given,  the  other  two  may  be  easily  found 
1st,  the  first  term  ;  2d,  the  last  term  ;  3d,  the  number  of  terms  ; 
4th,  the  common  difference  ;  5th,  the  sum  of  all  the  terms. 

The  first  term,  the  last  term,  and  the  number  of  terms  being  given t, 
to  find  the  common  difference, 

RULE    I. 

Divide  the  difference  of  the  extremes  by  the  number  of  terms, 
less  1,  and  the  quotient  will  be  the  common  difference  sought. 

1 .  The  extremes  are  3  and  39,  and  the  number  of  terms  is 
19  ;  what  is  the  common  dfference  ? 
Extremes. 
39-.3=36~19-l  =  18,  or  18)36(2  Ans. 
'2.  A  man  had  10  sons  whose  several  ages  differed  alike; 
the  youngest  was  3  years  old,  and  the  eldest  48  ;  what  was  the 
common  difference  of  their  ages  ?  Ans.  5  years. 

The  first  term,  the  last  term,  and  the  number  of  terms ^  being  given, 
to  find  the  sum  of  all  the  terms. 

RULE    II. 

Multiply  the  sum  of  the  extremes  by  the  number  of  terms, 
and  half  the  product  will  be  the  answer. 

3.  A  lady  purchased  19  yards  of  riband,  for  which  she  gave 
Ic.  for  the  first  yard,  3c.  for  the  second,  and  5c.  for  the  third 
yard,  increasing  2c.  per  yard  ;  required  the  cost  ? 

19  less  1  =  18  then  1  +  37=38 

com.  diflf.  2  X  X 19  no.  of  terms. 

36                               2)722 
1st  term,  +1  

—        half  product,  D3.61  Ans. 
last  term,    37 

4.  If  100  stones  were  laid  two  yards  distant  from  each  other, 
in  a  right  line,  and  a  basket  placed  two  yards  from  the  first 
stone  ;  what  distance  would  a  person  travel,  to  gather  them 
singly  into  a  basket?  Ans.  11  m.,  3  fur.  180  yds. 

Note. — In  this  question,  there  being  1760  yards  in  a  mile, 
and  the  man  returning  with  each  stone  to  the  basket,  his  travel 
will  be  doubled. 


214  :  ARITHMETICAL    PROGRESSION. 

Given  the  extremes  and  the  common  difference,  to  find  the  numhet 
of  terms. 

RULE    HI. 

Divide  the  difference  of  the  extremes  by  the  common  differ- 
ence, and  the  quotient,  increased  by  1,  will  be  the  number  of 
terms  required. 

5.  The  extremes  are  3  and  39,  and  the  common  difference  2  ; 
what  is  the  number  of  terms  ? 

39  —  3  =  36—2  com.  diff.  18  quot.  +  l  =  19.  Ans, 

6.  A  man  going  a  journey  travelled  the  first  day  7  miles,  the 
last  day  51  miles,  and  each  day  increased  his  journey  by  4  miles  ; 
how  many  days  did  he  travel,  and  how  far?  Ans.  12  days ;  348  m. 

The  extremes  and  common  difference  given,  to  find  the  sum  of  all 
the  series. 


Multiply  the  sum  of  the  extremes  by  their  difference,  increased 
by  the  common  difference,  and  the  product,  divided  by  twice 
the  common  difference,  will  give  the  sum. 

7.  If  the  extremes  are  3  and  39,  and  the  common  difference 
2,  what  is  the  sum  of  the  series?  39  +  3=42  sum  of  the  ex- 
tremes ;  39  —  3  =  36,  difference  of  extremes  ;  36  +  2  =  38,  differ- 
ence of  extremes  increased  by  the  common  divisor;  42x38  = 
1596-i-4,  twice  the  common  difference  =399.  Ans. 

The  extremes  and  sum  of  the  series  given,  to  find  the  number  of 

terms. 


Twice  the  sum  of  the  series,  divided  by  the  sum  of  the  ex- 
tremes, will  give  the  number  of  terms. 

8.  Let  the  extremes  be  3  and  39,  and  the  sum  of  the  series 
399  ;  what  is  the  number  of  terms  ?  Sum  of  the  series  =  399 
X 2  =  798  ;  then  sum  of  the  extremes  =39  +  3=42  :  798-r-42 
=  19.  Ans. 

The  extremes  and  the  sum  of  the  series  given,  to  find  the  common 
difference, 

RULE    VI. 

Divide  the  product  of  the  sum  antj  the  difference  of  the  ex- 
tremes by  the  difference  of  twice  the  sum  of  the  series  and  the 
sum  of  the  extremes,  and  the  quotient  will  be  the  common  dif- 
ference. 


ARITHMETICAL    PROGRESSION.  215 

9.  Let  the  extremes  be  3  and  39,  and  the  sum  399 ;  what  is 
whe  common  difference?  Sum. — Extremes  =  39-f  3=42  ;  dif- 
ference of  extremes=39— 3=36  ;  42x36=1512  ;  then  399 
X2-42=756)1512(2  Ans. 

The  first  term,  the  common  difference,  and  the  number  of  terms 
given,  to  find  the  last  term, 

RULE    VII. 

The  number  of  terms,  less  1,  muhiplied  by  the  common  dif- 
ference, and  the  first  term  added  to  the  product,  will  give  the 
last  term. 

10.  A  man  bought  100  yards  of  cloth,  giving  4c.  for  the  first 
yard,  7c.  for  the  second,  10c.  for  the  third,  and  so  on,  with  a 
common  difference  of  3c.  ;  what  was  the  cost  of  the  last  yard  ? 

No.  of  terms,  100  —  1  =  99x3,  com.  difr.=297+4=the  first 
term  =  3.0lD.  Ans. 

11.  A  man  travelled  20  miles  the  first  day,  24  the  second, 
and  so  on,  increasing  4  miles  every  day ;  how  far  did  he  travel 
the  12th  day  ?  Ans.  64  m. 

Note. — The  great  variety  of  cases  in  arithmetical  progres- 
sion will  not  permit  more  space  to  be  occupied  in  this  work ; 
such  questions  as  usually  occur  may  be  solved  by  the  above  rules. 

12.  A  man  had  8  sons,  whose  ages  differed  alike  ;  the  young- 
est was  10  years  old,  and  the  eldest  45;  what  was  the  com- 
mon difference  of  their  ages  ?  Ans.  5  years. 

13.  A  man  is  to  travel  from  New  York  to  a  certain  place  in 
19  days,  and  to  go  but  six  miles  the  first  day,  increasing  every 
day  by  an  equal  excess,  so  that  the  last  day's  journey  may  be  60 
miles  ;  what  is  the  common  difference,  and  distance  of  the 
journey  ?  Ans.  com.  diff.  3  miles  ;  distance  627  miles. 

14.  It  is  required  to  know  how  many  times  the  hammer  of  a 
clock  would  strike  in  a  week,  or  168  hours,  provided  it  increases 
1  at  each  hour  ?  Ans.  14196. 

15.  What  will  Dl  at  6  per  cent,  amount  to  in  20  years,  at 
simple  interest  ?  Ans.  D2.20. 

16.  How  many  times  does  a  regular  clock  strike  in  12  hours  ? 

Ans.  78. 

17.  A  man  bought  100  oxen,  and  gave  for  the  first  ox  Dl  ; 
for  the  second,  D2  ;  for  the  third,  D3  ;  and  so  on  to  the  last ; 
how  much  did  they  come  to  1  Ans.  5050. 

REVIEW. 

What  is  Arithmetical  Progression  1  Name  the  five  things 
that  should  be  particularly  attended  to  in  this  rule  ?  How  do  you 


216  GEOMETRICAL  PROGRESSION. 

form  an  arithmetical  series  ?  What  is  the  common  difference  ? 
What  is  an  ascending  series  ?  What  is  a  descending  series  ? 
What  are  the  several  numbers  called  ?  What  are  the  first  and 
last  terms  called  ?  Ans.  Extremes,  and  the  intermediate  terms 
are  called  the  means.  How  do  you  find  the  common  difference, 
when  you  know  the  two  extremes  and  number  of  terms  ?  What 
is  lule  1st?  2d  ?   &c. 

18.  If  a  piece  of  land,  60  rods  in  length,  be  20  rods  wide  at 
one  end,  and  at  the  other  terminate  in  an  angle  or  point,  what 
number  of  square  rods  does  it  contain  ?  Ans.  600. 


GEOMETRICAL  PROGRESSION. 

Geometrical  Progression  is  a  series  of  numbers  increasing 
or  decreasing  by  a  common  ratio  ;  thus,  2,  4,  8, 16,  32,  increased 
by  the  multiplier  or  ratio  2,  and  32,  16,  8,  4,  2,  decrease  con- 
tinually by  the  divisor  2,  &;c.  In  this  rule  there  are  five  denom- 
inations, any  three  of  which  being  given,  the  other  two  may  be 
found  :  1st,  the  first  term  ;  2d,  the  last  term  ;  3d,  the  number  of 
terms  ;  4th,  the  ratio  ;  5th,  the  sum  of  the  series.  The  ratio  is; 
the  multiplier  or  divisor  by  which  the  series  is  founded.  To 
raise  a  power  or  series  of  numbers  by  the  ratio,  we  place  the 
ratio  at  the  left  hand  for  the  first  power  ;  this  (first  power)  mul- 
tiplied by  the  ratio  (its  square)  gives  the  second  power,  the  sec* 
ond  by  the  ratio  gives  the  third,  and  so  on,  until  the  power  is 
one  number  less  than  the  first  term.  Let  2,  4,  8,  16,  32,  &c.y 
be  the  series  whose  ratio  is  2.  The  second  term  is  formed  by 
multiplying  the  first  term  by  the  ratio  2  ;  the  third  term  by  mul- 
tiplying the  second  by  the  ratio,  and  so  on.  The  series  may 
therefore  be  written  thus  :  2,  2x2,  2x2x2,  2x2x2x2,  2x 
2X2X2X2,  &c.,  or  thus:  2,  2x2^,  2x2^,  2x2^,  2x2^&c• 
Any  power  after  the  first  is  evidently  that  power  of  the  ratio 
whose  index  is  one  less  than  the  number  of  the  term  multiplied 
by  the  first  term ;  thus  the  third  term  is  2  x22,  the  4th  term  is 
2  X  2^,  and  the  8th  term  would  be  2  X  2^,  &c.  In  an  ascending 
series,  therefore,  multiply  the  first  term  by  that  power  of  the 
ratio  whose  index  is  one  less  than  the  number  of  the  term  sought, 
as  mentioned  above,  and  the  product  is  the  term  sought,  la  a 
descending  series,  as  243,  81,  27,  9,  3,  1,  whose  ratio  is  3,  and 
which  is  also  lx3^  lx3S  1x33,  Ix32,lx3\  1, 

1X3^ 
the  last  term  1  =  

3^ 


aEOMETRICAL    PROGRESSION.  217 

Hence  we  derive  the  following  general  rule,  the  correctness  of 
which  is  evident,  whatever  be  the  first  term,  or  ratio. 

RULE. 

Raise  the  ratio  to  the  power  whose  index  is  one  less  than  the 
number  of  terms  given,  which  multiply  by  the  first  term  ;  that 
product  is  the  last  term,  or  greater  extreme. 

2.  Multiply  the  last  term  by  the  ratio,  and  from  the  produc 
subtract  the  first  term,  and  divide  the  remainder  by  the  ratio 
less  one  ;  the  quotient  will  be  the  sum  of  the  series. 

Or,  raise  the  ratio  to  a  power  equal  to  the  number  of  terms  ; 
subtract  one  from  that  power  ;  multiply  the  remainder  by  the 
first  term  ;  divide  this  product  by  the  ratio,  less  one ;  the  quo- 
tient will  be  the  sum  of  a  geometrical  series. 

1.  The  first  term  is  3,  and  the  ratio  2  ;  what  is  the  6th  term? 
2X2X2X2X2  =  2^=32;  32  x3,  1st  term,  z=:96.  Ans, 

Having  given  the  ratio  and  the  two  extremes^  to  find  the  surn  of 
the  series. 


Subtract  the  less  extreme  from  the  greater;  divide  the  re- 
mainder by  one  less  than  the  ratio,  and  to  the  quotient  add  the 
greater  extreme ;  the  sum  will  be  the  sum  of  the  series. 

2.  The  first  term  is  3,  and  the  ratio  2,  and  the  last  term  192 ; 
what  is  the  sum  of  the  series  ?   192 —  3=:  189,  diff.  of  extremes, 

2  —  1  =  1)189(189;  then  189+ 192z=:381.  Ans. 

3.  If  the  first  term  be  2,  and  the  ratio  2,  what  is  the  13th 
term?    1,2,3,    4,    5x    5  x3==13,  or,  2x21^  (less,  1)  =  8192. 

2,  4,  8,  16,  32x32x8=8192.  Ans. 

4.  If  the  first  term  be  5,  and  the  ratio  3,  what  is  the  7th 
term  ? 

0,  1,  2,  3,+2  +1  =  6 = indices  to  6th  t'm  beyond  1st  or  7th. 
5,  15,45,  135,X45X  15=91125  dividend. 

The  number  of  terms  multiplied  is  three,  namely  :  135x45 
X  15,  and  3  —  1=2,  is  the  power  to  which  the  term  5  is  to  be 
raised;  but  the  second  power  of  5  is  5x5=25,  therefore, 
91125-^25  =  3645=7,  term  required. 

Note. — The  following  examples  will  embrace  the  general  op- 
eration of  the  several  rules  of  geometrical  progression,  sufficient 
for  common  use. 

5.  A.  purchased  24  yards  of  sheeting,  for  wnich  he  paid  2c. 
for  the  first  yard,  4c.  for  the  second,  8c.  for  the  third,  <fec., 
increasing  in  a  duplicate  proportion  ;  required  the  amount. 

19 


218 


GEOMETRICAL    PROGRESSION. 

Thus,         12  3     4  indices. 

2  4  8  16  leading  terms 
16X 

256= 8th  term. 
256  X 


65536  =  16th  term. 
256  X 


16777216=24th  term. 

2  X  ratio,  raultiplicL 


33554432 

2— ratio,  subtracted. 


2-^1  =  1)335544.30  Ans.  in  dollars  and  cents. 
6.  What  debt  can  be  discharged  in  a  year,  by  paying  1  cen* 
the  first  month,  10  cents  the  second,  and  so  on,  each  mon^h  ii' 
a  tenfold  proportion  1 

12  3  4  5  6 

10  X 100  X 1000  X 10000  X 100000  X 1000000 

1000000 


1000000000000 

1" 

999999999999 
IX 


10-1  =  9)999999999999 


Dllllllllll.il  Ans. 

7.  What  will  one  cent  amount  to,  if  you  double  it  every  year 
for  21  years  1  Ans.  D20971.51. 

8.  The  first  term  of  a  series  having  10  terms,  is  4,  and  the 
ratio  3,  what  is  the  last  term  ?  Ans.  78732 

9.  What  is  the  last  term  of  a  series  having  18  terms,  the 
first  of  which  is  3,  and  the  ratio  3  ?  Ans.  387420489. 

10.  A  man  was  to  travel  to  a  certain  place  in  4  days,  and  to 
travel  at  whatever  rate  he  pleased  ;  the  first  day  he  went  2  miles, 
the  second  6  miles,  and  so  on  to  the  last,  in  a  threefold  ratio ; 
how  far  did  he  travel  the  last  day,  and  how  far  in  all  ? 

Ans,  last  day,  54  miles ;  in  all,  80  ndlo^ 


PERMUTATION.  219 

11.  Sold  14  pairs  of  stockings,  the  first  at  4  cents,  the  second 
at  12  cents,  and  so  on  in  a  geometrical  progression  ;  what  did 
the  last  pair  bring  him,  and  what  did  the  whole  bring  him  ? 

Ans.  last,  D63772.92  ;  whole,  D95659.36. 

12.  If  our  ancestors  who  landed  at  Plymouth,  A.  D.  1620, 
being  101  in  number,  had  increased  so  as  to  double  their  num- 
ber in  every  20  years,  how  great  would  have  been  their  popu- 
lation at  the  end  of  1840  ?  Ans.  206747. 
•  13.  A  sum  of  money  is  to  be  divided  among  10  persons  ;  the 
first  to  have  DIO,  the  second,  D30,  and  so  on,  m  a  threefold 
proportion;  what  will  the  last  have  ?                    Ans    D196830. 

14.  A  man  bought  a  horse,  and  by  agreement  was  to  give  Ic. 
for  the  first  nail,  2c.  for  the  second,  4c.  for  the  third,  &c.  ;  there 
were  4  shoes,  and  8  nails  in  each  shoe  ;  what  was  the  cost  of 
the  horse  ?  Ans.  D42949672.95. 


What  is  Geometrical  Progression  ?  How  do  you  form  a 
geometrical  progression  ?  What  is  the  common  ratio  ?  What 
is  the  difl^erence  between  arithmetical  and  geometrical  progres- 
sion ?  What  is  an  ascending  series  ?  What  is  a  descending 
series  ?  What  are  the  several  numbers  called  ?  In  every  geo- 
metrical progression  how  many  things  are  to  be  considered  ? 
What  are  they  ?  What  are  the  first  and  last  terms  called  ? 
What  are  the  intermediate  terms  called  ? 

15.  If  the  ratio  be  4,  the  number  of  terms  6,  and  the  greatest 
term  3072  ;  what  is  the  sum  of  the  series  ?  ^ns.  4095. 

Divide  the  last  term  by  the  5th  power  of  the  ratio,  &c 


PERMUTATION. 

Permutation  is  the  method  of  finding  how  many  diflferent 
ways  any  number  of  things  may  be  changed.  Thus  take  the 
first  three  letters  of  the  alphabet,  a  b  c  ;  they  will  admit  of  six 
changes,  a  b  c,  a  c  b,  b  a  c,  b  c  a,  c  b  a,  cab,  and  so  on, 
according  to  the  given  number  of  terms. 

RULE. 

Multiply  all  the  terms  of  the  natural  series  constantly  from  1, 
or  unity,  to  the  given  number,  inclusive  ;  the  last  product  will 
be  the  number  of  changes  required. 


220  COMBINATION. 

1.  In  how  many  different  positions  can  five  persons  be  placed 
at  a  table?  Thus:    1x2x3x4x5  =  120.  Ans, 

2.  What  time  will  it  require  for  8  persons  to  seat  themselves 
differently  every  day  at  dinner  ?        Ans.  110  years,  142^  days. 

3.  How  many  variations  can  be  made  of  the  English  alpha- 
bet, it  consisting  of  26  letters  ? 

^;25.  403291461126605635584000000. 

4.  How  many  changes  may  be  rung  on  15  bells  ;  and  in  what 
time  may  they  be  rung,  allowing  three  seconds  to  every  round  ? 

Ans.  1307674368000  changes;  3923023104000  seconds. 

5.  How  many  variations  may  there  be  in  the  position  of  the 
nine  digits  ?  Ans.  362880. 

6.  A  man  bought  25  cows,  agreeing  to  pay  for  them  1  cent 
for  every  different  order  in  which  they  could  all  be  placed  ;  how 
much  did  the  cows  cost  him  ? 

Ans.  D155112100433309859840000. 

7.  Christ's  church,  in  Boston,  has  8  bells ;  how  many 
changes  may  be  rung  upon  them.  Ans.  40320. 


COxMBINATION. 


Combination  teaches  how  many  different  ways  a  less  num- 
ber of  things  may  be  combined  out  of  a  larger  ;  thus  out  of  the 
letters  abed,  are  six  different  combinations  of  two,  namely, 
a  b,  a  c,  a  d,  d  c,  d  b,  b  c. 

Thus:  4x3  =  12;  1  x2=2)12(  =  6.  .4^^. 

RULE. 

Take  a  series,  proceeding  from  and  increasing  by  a  unit,  up 
to  the  number  to  be  combined  ;  then  take  a  series  of  as  many 
places,  decreasing  by  unity,  from  the  number  out  of  which  the 
combinations  are  to  be  made  ;  multiply  the  first  continually  for 
a  divisor,  and  the  other  for  a  dividend ;  the  quotient  will  be  the 
nswer. 

1.  How  many  combinations  of  five  letters  in  ten? 

Thus:  10x9x8x7x6  =  30240,  dividend;  1x2x3x4x5 
=  120,  divisor.  Ans.  252. 

2.  How  many  combinations  of  ten  figures  may  be  made  out 
of  twenty?        "  Ans.  184756. 

3.  How  many  combinations  may  be  made  of  seven  letters  out 
of  twelve.  Ans.  792 


COMPOUND    INTEREST  22i 

4.  How  many  combinations  can  be  made  of  six  letters  out  of 
the  24  letters  of  the  alphabet?  Ans.  134596. 

REVIEW. 

What  is  Permutation?  What  is  the  rule?  What  is  com- 
bination ?     What  is  the  rule '' 

5.  How  many  changes  may  be  rung  with  4  bells  out  of  8  ? 

Ans.  1680. 

6.  How  many  variations  may  be  made  of  the  letters  in  the 
word  Zaphnathpaaneah(15)?  2x6x1x2x120=2880  divi- 
sor. Ans.  454053600. 

7.  How  many  different  numbers  can  be  made  of  the  follow- 
ing figures  :   1223334444  ?  Ans.  12600. 


COMPOUND  INTEREST,  BY  DECIMALS. 

Compound  Interest  is  that  which  arises  from  interest  and 
principal  added  together  annually,  as  the  interest  becomes  due, 
by  the  continued  multiplication  of  the  new  principal  by  the  ratio 
or  rate  per  cent. ;  thus,  if  I  owe  A.  DlOO,  payable  on  demand, 
and  neglect  to  pay  either  interest  or  principal  for  several  years, 
he  would  be  justified  in  adding  the  interest  to  the  principal  an- 
nually, and  computing  the  interest  on  this  amount:  DlOO +  6 
=  106,  first  year;  then  Dl06x6=6.36  interest  second  year, 
which  D6.004-6.36D.  =  12.36  interest  or  Dl  12.36  amount  for 
the  second  year  ;  Dl  12.36  X  6=6.74  interest  for  the  third  year, 
which  D6. 00  +  6. 36  +  6.74  =  19. 10  interest  for  the  third  year,  or 
Dl  19.10.160  amount  at  compound  interest  for  three  years  at  6 
per  cent.  In  many  cases  it  is  considered  illegal  to  receive 
compound  interest,  therefore  it  is  seldom  computed ;  but  when 
a  note,  bond,  or  obligation,  is  given  with  a  credit  of  several  years, 
on  condition  that  the  interest  shall  be  paid  annually,  if  the  inter- 
est is  not  paid  until  the  obligation  becomes  due,  it  is  no  more 
than  just  and  right  that  compound  interest  should  be  paid,  as 
well  as  principal ;  for  the  person  having  the  use  of  the  princi- 
pal has  likewise  the  use  of  the  interest,  which  of  right  belongs 
to  another ;  therefore  interest  should  be  computed  accordingly. 

The  following  table  is  computed  by  the  continual  multiplica- 
tion of  1,  or  DlOO  by  the  ratio  as  above;  which  may  some- 
times be  found  useful  where  the  higher  powers  are  required,  or 
»o  test  the  accuracy  of  calculations. 

19* 


2Q2 


COMPOUND    INTEREST. 


A  tkible  showing  the  amount  of  Dl^  or  DlOO,  for  15  t^ears  at  4, 
5 J  6,  and  7  per  cent,  at  compound  interest. 


Years. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

1 

1.0400000 

1.050000 

1.060000 

1.070000 

3 

1.0816000 

1.102500 

1.123600 

1.144900 

3 

1.1248640 

1.157625 

1.191016 

1.225043 

4 

1.1698585 

1.215506 

1.262477 

1.310795 

5 

1.2166529 

1.276281 

1.338225 

1.402552 

6 

1.2653190 

1.340095 

1.418519 

1.500730 

7 

1.3159317 

1.407100 

1.503630 

1.605781 

8 

1.3685690 

1.477455 

1.593848 

1.718186 

9 

1.4233118 

1.551328 

1.689479 

1.838459 

10 

1.4802842 

1.628894 

1.790848 

1.967151 

11 

1.5394540 

1.710339 

1.898299 

2.104852 

12 

1.6010322 

1.795856 

2.012197 

2.252191 

13 

1.6650735 

1.885649 

2.132928 

2.409845 

14 

1.7316764 

1.979931 

2.260904 

2.578534 

15 

1.8009435 

2.078928 

2.396558 

2.759032 

TABLE    II. 


A  table  showing  the  amount  of  Dl  or  DlOO  for  one  year,  and 
for  quarters  of  a  year,  at  compound  interest,  and  simple  inter' 
est  for  one  month. 


rate  per  c. 

ratio. 

3quart'rs.  2quart'rs. 

1  quarter. 

S.int.  Imo, 

4 

1.04 

1.029852 

1.019804 

1.009853 

.003333 

4.5 

1.045 

1.033563 

1.022252 

1.011065 

.003750 

5 

J. 05 

1.037270 

1.024675 

1.012272 

.004167 

5.5 

1.055 

1.040973 

1.027132 

1.013475 

.004583 

6 

1.06 

1.044671 

1.029536 

1.014674 

.005000 

To  compute  interest  by  the  use  of  the  tables,  find  the  amount 
or  tabular  number,  under  the  rate  per  cent.,  and  opposite  the  num- 
ber of  years  ;  multiply  ^is  number  by  the  principal,  and  you 
have  the  amount,  from  which  subtract  the  principal,  and  the  re- 
mainder will  be  the  compound  interest.  If  quarterly  payments 
ere  required  at  compound  the  4th  root  of  the  ratio  • 


CX>MPOUND    INTERES'^.  2S^8 

for  half-yearly,  the  square  root ;  and  for  three  quarters,  the  prod- 
uct of  the  quarterly  and  half  yearly ;  thus,  the  tabular  number 
at  6  per  cent.,  will  be  for  one  quarter  1.014674  ;  for  two  quar- 
ters, 1.029536;  for  3  quarters,  1.044671,  at  6  per  cent.  Or 
find  the  powers  by  the  tables,  which  multiply  together,  and  the 
product  will  be  the  amount ;  thus,  for  4.25  years  at  6  per  cent, 
per  annum,  1.2624769  X  1.014674  =:1.2810024860306  amount 
of  DlOO,  (fee,  or  D28.10  interest  for  4A  years.  It  may  be  found 
very  nearly  by  finding  the  compound  interest  for  the  years,  and 
then  compute  the  simple  interest  on  that  amount  for  quarterly 
or  parts  of  a  year. 

Again,  ^1.06  =  1.014674  quarterly  amount:  and  ^1.06  = 
1.029536  half-yearly  amount:  then,  1.014674x1.029536  = 
1.044671:  then,  for  3^  years  at  6  per  cent.,  1.191016X 
]. 0114674  =  1.2184929,  amount  for  3}  yrs.  at  6  per  ct.  per  ann. 

1.  What  is  the  compound  interest  of  DlOO  for  4  years,  at 
6  per  cent,  per  annum  ?  By  the  table,  1.2624769  X  100  = 
1.262476900-1.00  =  D26.24,7,6900.  Ans, 

2.  What  is  the  amount  of  Dl50p  for  12  years  at  3.5  per  ct. 
per  annum?  Thus,  tabular  number,  1.5110686x1500  = 
D2266.60.29.     (The  tabular  number  is  not  in  the  table.) 

RULE    II. 

Multiply  the  principal  by  the  ratio,  which  will  give  the  simple 
interest  for  1  year ;  add  this  interest  to  the  principal  and  calcu- 
late as  before  for  the  second  year,  and  so  continue  up  to  the 
number  of  years  required,  and  the  last  product  will  be  the 
amount,  from  which  subtract  the  principal  and  the  remainder 
will  be  the  compound  interest. 

3.  What  is  the  compound  interest  of  DlOO  for  4  years  at  6  per 
cent,  per  annum  ?     Thus,  DlOO  principal, 

DlOO  x  6  ratio  =  6  interest  1st  year, 

106  amount  1st  year, 
106  X  6=  6.36  interest  2d  year, 


112.36  amount  2d  year. 
112.36  X6=t  6.7416  interest  3d  year. 

119.1016  amount  3d  year. 
119.1016x6=  7.146096  interest  4th  year. 

126.24.7.696  amount  4th  year. 
Principal     —100 

D26.24,7.696.  Ans. 


224  COMPOUND    INTEREST. 

4  What  is  the  compound  interest  of  D500  for  4  years  ax  6 
percent.?  Ans.  D131.23.8. 

5.  What  is  the  compound  interest  of  D700  for  7  years  at  5  pet 
cent.  ?  Ans.  D284.97. 

6.  What  is  the  compound  interest  of  D850  for  10  years  at  6 
per  cent.  ?  Ans.  D672.22.08. 

7.  What  is  the  compound  interest  of  Dl  100  for  15  years  at  5 
per  cent.  ?  Ans.  D1186.82.08. 

8.  What  is  the  compound  interest  of  D4400  for  20  years  at  6 
per  cent.  ?  Ans.  D971 1 .39.6. 

9.  What  is  the  compound  interest  of  D20000  for  13  years  at  6 
per  cent.  ?  Ans.  D22658.56 

10.  What  is  the  amount  of  D768  for  3 J  years  at  6  per  cent 
compound  interest  ?  Ans.  For  3  years,  1.1910160  X  1.014674  for 
i=:  12084929687840  amount  of  lOOfor  3-i  ;  then  x768D.princ. 
=  D928.12.2.  Ans. 

11.  What  is  the  difference  between  the  simple  interest  of 
D200  for  3  years,  and  the  compound  interest  for  the  same  time  ? 

Ans,  D2.2033^. 

When  the  amount,  rate,  and  time,  are  given,  to  find  the  principal 

RULE    III. 

Divide  the  amount  by  the  amount  of  1  or  D 100  for  the  given 
time,  and  the  quotient  will  be  the  principal. 

Or  if  you  multiply  the  present  value  of  1  or  DlOO  for  the 
given  number  of  years,  at  the  given  nate  per  cent,  by  the  amount 
the  product  will  be  the  principal  or  present  worth. 

12.  What  principal  must  be  put  to  interest  6  years,  at  5^  pe ,' 
cent,  per  annum,  to  amount  to  D689.4214033809453125  ? 

Thus,  1.3788426)689.4214033809453125(500D.  Ans. 

Note. — We  have  a  variety  of  cases  in   Compound  Interes  , 
and  Annuities,  which,  for  want  of  room,  are  necessarily  omitted, 
a  sufficient  number  will  be  given  for  all  practical  purposes. 

REVIEW. 

What  is  Compound  Interest  ?  How  can  you  compute  inter- 
est by  the  use  of  the  tables  ?  What  is  rule  2  ?  How  can  you 
find  the  principal  1  In  what  time  will  any  sum  double  at  6  per 
cent,  simple  interest  ?  Ans.  16  years,  8  months.  In  what  time 
at  compound  interest?  Ans.  11  years,  8  months,  22  days 
When  is  it  lawful  to  compute  compound  interest  ? 

13.  What  is  the  compound  interest  of  the  following  sums 
D1200  for  5  years,  and  D480  for  7  years,  at  7  per  cent.  ? 


ANNUITIES.  32§ 

ANNUITIES. 

An  Annuity  is  a  sum  of  money  payable  at  regular  periods, 
for  a  certain  time,  or  for  ever.  Annuities  sometimes  depend  on 
some  contingency,  as  the  life  or  death  of  a  person,  and  the  an 
nuity  is  then  said  to  be  contingent.  Sometimes  annuities  are 
not  to  commence  till  a  certain  number  of  years  have  elapsed, 
and  the  annuities  are  then  said  to  be  in  reversion.  The  annuity 
is  said  to  be  in  arrears  when  the  debtor  keeps  it  beyond  the 
time  of  payment.  The  present  worth  of  an  annuity  is  such  a 
sum,  as  being  now  put  out  at  interest  would  exactly  pay  the  an- 
nuity, as  it  becomes  due,  and  is  the  sum  which  must  be  given 
for  the  annuity  if  it  be  paid  at  its  commencement.  The  amount 
is  the  sum  of  the  annuities  for  the  time  it  had  been  forborne, 
with  the  interest  due  on  each. 

To  Jind  the  amount  of  an  annuity,  at  simple  interest. 

RULE    I. 

Multiply  the  natural  series  of  numbers,  1,  2,  3,  4,  &c.,  to  the 
number  of  years,  less  1,  by  the  interest  of  the  annuity  for  one 
year,  and  the  product  will  be  the  interest  which  is  due  on  the 
annuity.  Multiply  the  annuity  by  the  time,  and  the  sum  of  the 
two  products  will  be  the  amount. 

1.  What  is  the  amount  of  an  annuity  of  DlOO,  for  4  years, 
computing  interest  at  6  per  cent.  ?  Thus  :  l-f-2+3  =  6,  sum  of 
the  natural  series  to  the  number  of  years,  less  1.  D6  interest 
of  annuity  for  one  year  ;  then  6x6  =  D36,  the  whole  interest  - 
100x4=D400  +  36=436,  amount.  Ans.  (See  note  at  close 
of  annuities.) 

2.  If  a  pension  of  D20  be  continued  unpaid  for  6  years,  what 
is  its  amount  at  6  and  7  per  cent.  ?  At  6  per  cent.,  1)138  ;  at  7 
per  cent.,  D141.  Ans. 

REMARKS. 

It  is  plain  that  upon  the  first  year's  annuity  there  will  be  duo 
so  many  years'  interest  as  the  given  number  of  years,  less  1, 
and  gradually  one  year  less  upon  each  such  succeeding  year,  to 
that  preceding  the  last,  which  has  but  one  year's  interest,  and 
the  last  bears  none.  There  is,  therefore,  due  in  the  whole  as 
many  years'  interest  of  the  annuity  as  the  sum  of  the  series,  1, 
2,  3,  &;c.,  to  the  number  of  years,  diminished  one.  It  is  evi- 
dent then,  that  the  whole  interest  due  must  equal  this  sum  of  the 
natural  series,  multiplied  by  the  interest  for  one  year,  and  that 
the  amount  will  be  all  the  anrmities,  or  the  product  of  the  an- 
nuity and  time  added  to  the  whole  interest ;  and  this  is  the  rule.- 


Z26 


ANNUITIES. 


The  annuity  J  time,  and  ratio  given,  to  find  the  amount  at  com" 
pound  interest. 

RULE    II. 

1.  Make  1  the  first  term  of  a  geometrical  progression,  and 
the  amount  of  Dl,  and  tlie  given  rate  per  cent,  the  ratio. 

2.  Carry  the  series  to  as  many  terms  as  the  number  of  years, 
and  find  the  sum. 

3.  Multiply  the  sum  thus  found  by  the  given  annuity,  and  the 
product  will  be  the  amount  sought.  Or,  multiply  the  amount  of 
Dl  for  one  year,  at  the  given  rate  per  cent.,  into  itself  as  many 
times  as  there  are  years  given;  from  the  product  subtract  one, 
then  divide  the  remainder  of  the  interest  of  Dl  for  1  year,  av 
its  given  rate  per  cent.,  and  multiply  the  quotient  by  the  annuity 
for  the  amount  required. 


TABLE 

I.                     1 

TABLE    II. 

A  table  showing  the  amount  of 

A    table  showing   the    present 

Dl    or  DlOO,  from  1   to  15 

worth  of  D I  or  BlOO,  from  1 

years,  at  5  and  6  per  cent. 

to  15  years,  at  5  and  6 percent. 

Years. 

5  per  cent. 

6  per  cent. 

5  per  cent. 

6  per  cent. 

Years 

1 

1. 

1. 

0.95238 

0.94339 

1 

2 

2.05 

2.06 

1.86941 

1.83339 

2 

3 

3.1525 

3.1836 

2.72325 

2.67301 

3 

4 

4.310125 

4.374616 

3.54595 

3.46510 

4 

5 

5.525631 

5.637093 

4.32988 

4.21236 

5 

6 

6.801913 

6.975318 

5.07569 

4.91732 

6 

7 

8.142008 

8.393837 

5.78637 

5.58338 

7 

8 

9.549109 

9.897467 

6.46321 

6.20979 

8 

9 

11.026564 

11.491315 

7.10782 

6.80169 

9 

10 

12.577892 

13.180794 

7.72173 

7.36009 

10 

11 

14.206787 

14.971642 

8.30640 

7.88687 

11 

12 

15.917126 

16.869940 

8.86325 

8.38384 

12 

13 

17.712983 

18.882132 

9.39357 

8.85668 

13 

14 

19.598632 

21.015064 

9.89864 

9.29498 

14 

15 

21.785635 

23.275968 

10.37965 

9.71225 

15 

TABLE    III. 

ratio. 

half  yearly. 

quarterly. 

4 

1.009902 

1.014877 

4.5 

1.011126 

1.016720 

5 

1.012348 

1.018550 

5.5 

1.013567 

1.020395 

6 

1.014781 

1.022257 

The  construction  of  table 
3  is  from  an  algebraic 
theorem,  which  may  be  in 
words,  thus  :  for  half-year- 
ly payments,  take  a  unit 
from  the  ratio,  and  from 
the  square  root  of  the  ratio ; 
half  the  quotient  of  the  firsr 


ANNUITIES.  22*^ 

remainder,  divided  by  the  latter  will  be  the  tabular  number ;  for 
quarterly  payments,  use  the  4th  root,  and  take  one  quarter  of  the 
quotient. 

Table  1st  is  calculated  thus:  take  the  first  year's  amount, 
which  is  Dl,  multiply  it  by  1.06-|- 1=2.06,  second  year's 
amount,  which  also  multiply  by  1. 06+1  =  3. 1836,  third  year's 
amount,  &c.,  at  6  per  cent.,  and  in  this  manner  calculate  the 
other  tables. 

RULE    III. 

Multiply  the  tabular  number  under  the  rate,  and  opposite  to 
the  time,  by  the  annuity,  and  the  product  will  be  the  amount. 

3.  What  will  an  annuity  of  D60  per  annum  amount  to  in  20 
years,  allowing  6  per  cent.,  compound  interest  ? 

Thus;  36.785592  X  D60  ann.:rrD2207.13.5520.  Ans, 

4.  What  will  an  annuity  of  D60  per  annum,  payable  yearly, 
amount  to,  in  4  years,  at  6  per  cent.  ?     Thus  :   1  + 1. 06 -f  1.062 
+  1.063  =  4.374616,  sum,  xD60,  ann.  =D262.47.696.  Ans. 
(Rule  2.) 

5.  What  will  a  pension  of  D75  per  annum,  payable  yearly, 
umcunt  to  in  9  years,  at  5  per  cent.,  compound  interest  ? 

Ans.  D826.99.2^V 

The  annuitr/,  time,  and  rate  given,  to  find  the  present  worth, 

RULE    IV. 

Divide  the  annuity  by  the  amount  of  Dl  for  1  year,  and  the 
quotient  will  be  the  present  worth  of  one  year's  annuity. 

2.  Divide  the  annuity  by  the  square  of  the  ratio,  and  the  quo- 
tient will  be  the  present  worth  for  2  years. 

3.  In  like  manner,  find  the  present  worth  of  each  year  by  it- 
self, and  the  sum  of  all  these  will  be  the  present  value  of  the 
annuity. 

6.  What  ready  money  will  purchase  an  annuity  of  D60,  to 
continue  4  years,  at  6  per  cent.,  compound  interest  ? 

ratio    =  1.06)60.00000(56.603=present  worth  1  year, 

ratio  2=  1.1236)60.00000(53.399  "  2  years, 

ratio  3=      1.191016)60.00000(50.377  "  3  years, 

ratio  4=  1.26247696)60.00000(47.525  «  4  years. 


D207.904  Ans. 
(See  table,  compound  interest.) 

OR,    BY    TABLE    2. 

Multiply  the  number  under  the  rate,  and  opposite  the  time,  in 


228        PERPETUlriES  AT  COMPOUND  INTEREST. 

the  table,  by  the  annuity,  the  product  will  be  the  present  worth 
for  yearly  payments. 

When  the  payments  are  to  be  made  quarterly,  or  half-yearly, 
the  present  worth,  found  as  above,  must  be  multiplied  by  the 
proper  number  in  table  3d. 
Thus  :  question  6,  tabular  No.  3,  46510xD60  =  207.906.  Ans. 

7.  What  is  the  present  worth  of  an  annuity  of  D60  per  an- 
num, for  20  years,  at  6  per  cent.  ?  Ans,  D688. 19.50. 

8.  What  is  the  present  worth  of  D75  per  annum,  for  7  years, 
at  5  per  cent.?  Thus  :  5.78637x75  =  0433.97.775.  Ans. 

Table  2d  is  thus  made  :  Divide  Dl  by  1.06  =  .94339  the  pres- 
ent worth  of  the  first  year,  which,  divided  by  1.06  is  equal  to 
.88999,  which  added  to  the  first  year's  present  worth  is  = 
1.83339,  the  second  year's  present  worth;  then  .88999  divided 
by  1.06,  and  the  quotient  added  to  1.83339  gives  2.67301  for 
the  3d  year's  present  worth,  &c. 

Annuities  in  reversion  at  compound  interest. 

RULE  V. 

Take  two  numbers  under  the  given  rate  in  table  2d,  that 
stand  opposite  the  sum  of  the  two  given  times,  and  the  number 
opposite  the  time  when  the  annui.v  is  to  commence,  or  time  of 
reversion,  and  multiply  their  difference  by  the  annuity  for  the 
present  worth. 

9.  The  reversion  of  a  freehold  estate  of  D60  per  annum,  for 
4  years,  to  commence  2  years  hence  ;  what  is  the  present  worth, 
allowing  4  per  cent,  for  present  payment  ?  Thus  :  tab.  no. 
5.24214-1.88609=3.35605  xD60  =  D201.36.300.  Ans. 

10.  What  is  the  present  worth  of  a  reversion  of  a  lease  for 
D120  per  annum,  to  continue  9  years,  but  not  to  commence  till 
the  end  of  4  years,  at  4  per  cent.,  to  the  purchaser  ? 

Thus:  9.98565  — 3.62989=6. 35576 X120=D762.69.1.  Ans. 


PERPETUITIES  AT  COMPOUND  INTEREST. 

Perpetuities  are  such  annuities  as  continue  for  ever. 
The  annuity  and  rate  given,  to  find  the  present  worth. 

RULE    VI. 

Multiply  the  amount  of  Dl  for  one  year,  at  the  given  rate  per 
cent.,  involved  to  the  time  of  reversion,  by  the  ratio,  for  a  divi- 
sor, by  which  divide  the  yearly  payments ;  the  quotient  will  be 
the  answer. 


ANNUITIES.  229 

11.  Suppose  a  freehold  estate  of  D140  per  annum,  to  com- 
«nence  three  years  hence,  is  to  be  sold  ;  what  is  it  worth,  allow 
in^  the  purchaser  7  per  cent.  ?  Thus,  1 .07  X  1 .07  X  1 .07  X  .07= 
.08575301;  then  Dl40^08575301  =  1632.59.5D.  Ans. 

12.  What  is  an  estate  of  D260  per  annum,  to  continue  foi 
ever,  worth  in  present  money,  allowing  6  per  cent,  to  the  pur 
chaser?  D4333.33.3.  Ans, 

7\    find  the  present  worth  of  a  freehold  estate,  or  an  annuity  to 
continue  for  ever  at  compound  interest, 

RULE    VII. 

As  the  rate  per  cent,  is  to  DlOO,  so  is  the  yearly  rent  to  the 
value  required. 

13.  What  is  the  worth  of  a  freehold  estate  of  D40  per  an- 
num, allowing  5  per  cent,  to  the  purchaser  1 

Thus,  D5  :  100  ::  40  :  800.  Ans, 

14.  What  is  the  amount  of  an  annuity  of  D 180  for  9  years  at 
5  per  cent.  ?*  Ans.  D1984.781520. 

15.  What  will  an  annuity  of  D200  amount  to  in  5  years,  to 
be  paid  by  half-yearly  payments  at  6  per  cent.  ?  Thus,  5.637093 
X200=1127.4186x  1.014781  =  1144.08.2D.  Ans. 

16.  Required  the  amount  of  an  annuity  of  D 150  for  10  years 
at  5  per  cent.  ?  Ans.  1886.68.3. 

17.  If  a  salary  of  DlOO  per  annum,  to  be  paid  yearly,  be  for- 
borne 5  yrs.  at  6  per  cent.,  what  is  the  amount  ?  Ans.  D563.70.9. 

18.  What  sum  of  ready  money  will  buy  an  annuity  of  D300, 
to  continue  10  years,  at  6  per  cent. 

10  years=::7.36008x 300=2208.0240.  Ans, 

19.  What  salary,  to  continue  10  years,  will  D2208.024  buy? 
This  example  is  the  reverse  of  the  one  above,  consequently 
D2208.024-^7.36008=300.  Ans, 

20.  If  the  annual  rent  of  a  house,  which  is  D150,  remain  in 
arrears  for  3  years  at  6  per  cent.,  what  will  be  the  amount  dua 
for  that  time  ?  Ans,  D477.54. 

Note. — From  the  nature  of  an  annuity,  as  explained  in  the 
proof  of  the  rule  in  annuities  at  simple  interest,  there  is  due  one 
year's  interest  less  than  the  number  of  years  the  annuity  has 
been  continued.  By  the  rule  and  examples  in  compound  inter- 
est, the  amount  of  Dl  at  the  given  rate  is  equal  to  that  power  of 
the  amount  for  one  year,  which  is  indicated  by  the  number  of 
years.  This  amount  is  obtained  for  one  less  than  the  number- 
of  years  by  forming  the  geometrical  series  as  directed  in  the  rule, 
or  beginning  with  unity ;  thus,  in  question  1  (annuities)  the  series 

*  See  preceding  rules. 
80 


230  DUODECIMALS. 

is  1,  1.06,  1.062,  1.063,  and  the  last  term  is  the  amount  of  Dl 
for  1  less  than  4,  the  number  of  years.  The  sum  of  this  series 
is  the  amount  at  compound  interest,  of  an  annuity  of  Dl  for  4 
years.  The  amount  of  any  other  annuity,  for  the  same  time  and 
rate,  will  be  as  much  greater  or  less  as  the  annuity  is  greater 
or  less  than  Dl  ;  that  is,  the  amount  of  the  annuity  of  Dl  must 
be  multiplied  by  the  annuity  to  obtain  its  amount,  <fec. 

REVIEW. 

What  is  an  annuity  ?  How  will  you  find  the  amount  of  an 
annuity  at  simple  interest  ?  How  will  you  find  the  amount  of 
an  annuity  at  compound  interest  ?  What  is  the  rule  ?  What 
will  you  do  when  the  payments  are  half-yearly,  or  quarterly  ? 
How  is  table  1st  calculated?  How  will  you  find  the  present 
worth  of  an  annuity  ?  What  is  the  rule  1  How  is  table  2d  cal- 
culated? When  quarterly  payments  are  required^  what  is  to  be 
done  ?  How  is  table  3  constructed  ?  What  can  you  say  of 
annuities  in  reversion  ?  What  is  the  rule  ?  What  name  is 
given  to  annuities  which  continue  for  ever  ?  What  is  the  rule 
for  perpetuities  ?     What  more  can  you  say  of  annuities  ? 


DUODECIMALS,  OR  CROSS  MULTIPLICATION. 

Duodecimals,  that  is,  numbering  by  12,  are  fractions  of  a 
foot,  or  of  an  inch,  or  parts  of  an  inch,  having  12  for  their  de- 
nominator. It  is  a  rule  in  general  use  with  artificers  in  casting 
up  the  contents  of  their  work,  and  an  excellent  and  useful  rule 
for  that  purpose.  Considering  a  foot  as  the  measuring  unit,  a 
prime  is  the  12th  part  of  a  foot,  a  second  the  12th  part  of  a 
prime,  &c.  It  is  to  be  observed,  that  in  measures  of  length, 
inches  are  primes,  but  in  superficial  measure  they  are  seconds. 
In  both,  a  prime  is  jL.  of  a  foot ;  but  -^^  ^^  ^  square  foot  is  a 
parallelogram,  a  foot  long  and  an  inch  broad.  The  12th  part  of 
this  is  a  square  inch,  which  is  yi-j  of  a  square  foot.  This 
method  of  multiplying  crosswise  is  not  confined  to  twelves,  but 
may  be  greatly  extended,  for  any  number,  whether  its  inferior 
denominations  decrease  from  the  integer  in  the  same  ratio  or 
not,  may  be  multiplied  crosswise.  Thus,  pounds  multiplied  by 
pounds  are  pounds,  pounds  multiplied  by  shillings  are  shillings, 
&c. ;  shillings  multiplied  by  shillings  are  twentieths  of  a  shil- 
ling ;  shillings  multiplied  by  pence  are  twentieths  of  a  penny, 
pence  multiplied  by  pence  are  240ths  of  a  penny,  <fec.    The  word 


DUODECIMALS. 


231 


duodecimal  is  derived  from  the  Latin  word  duodecim,  signifying 

twelve.     The  denominations  are — 

12  fourths  ''''  make       -         -         -         1  third       "' 
12  thirds  "  -         -         -         1  second   ^' 

12  seconds  "  -         -         .         1  inch        I. 

12  inches  "  -         -         -         1  foot         ft. 


ADDITION  OF  DUODECIMALS. 


Add  as  in  Compound  Addition,  carrying  1  for  each  12  to  th© 
next  denomination. 


EXAMPLES. 

(1)  ft. 

I. 

// 

/// 

nil 

(2) 

ft. 

I. 

// 

/// 

III/ 

14 

4 

3 

5 

6 

28 

4 

3 

7 

10 

85 

7 

8 

6 

6 

71 

7 

8 

4 

2 

56 

10 

5 

7 

9 

67 

11 

3 

7 

5 

43 

1 

6 

4 

3 

32 

0 

8 

4 

7 

87 

11 

10 

8 

5 

46 

3 

8 

11 

10 

48 

5 

2 

10 

11 

67 

10 

1 

4 

11 

ft.  336--5  — 1  — 7  — 4 


3.  Five  floors  in  a  certain  building  contain  each  1295  feet,  9 
inches,  8^^ ;  how  many  feet  in  all  ?  Ans.  6479ft.,  Oin.,  4'". 

4.  Several  boards  measure  as  follows,  namely :  27ft.,  3in. ; 
25ft.,  llin. ;  23ft.,  lOin. ;  20ft.,  9in. ;  20ft.,  6in. ;  and  18ft.,  5 
in. ;  what  number  of  feet  in  all?  Ans.  136ft.,  8in. 


SUBTRACTION  OF  DUODECIMALS. 


Work  as  in   Compound   Subtraction,  borrowing  12,   &c. 
when  necessary. 

(5)    ft.   in.  "      "'      ""   (6)   ft.   -   "      '"      "" 


in 
From  176  1 
Take  97  10 


6 

7 


10 
11 


Rem.  78   3   0  10   11 


m. 
3786  10   4 
987   8  11 


7 
9 


t^S8  DtrODECIMALS. 

7.  From  a  board  measuring  41ft.  7in.,  cut  19ft.  lOin. ;  wha 
remains  1  Ans.  21ft.  9in. 


MULTIPLICATION   OF   DUODECIMALS. 

RULE  I. 

1.  Under  tlie  multiplicand  write  the  corresponding  denomina- 
tions of  the  multiplier. 

2.  Multiply  each  term  of  the  multiplicand,  beginning  at  the 
lowest,  by  the  highest  denomination  in  the  multiplier,  and  write 
the  result  of  each  under  its  respective  term,  observing,  in  duo- 
decimals, to  carry  a  unit  for  every  12  from  each  lower  denom- 
ination to  its  next  superior,  and  for  other  numbers  accordingly. 

3.  In  the  same  manner,  multiply  all  the  multiplicand  by  the 
primes,  or  seconds,  denomination  in  the  multiplier,  and  set  the 
result  of  each  term  one  place  removed  to  the  right  of  those  in 
the  multiplicand. 

4.  Do  the  same  with  the  seconds  in  the  multiplier,  setting 
the  result  of  each  term  two  places  to  the  right  of  those  in  the 
multiplicand. 

5.  Proceed  in  like  manner  with  all  the  rest  of  the  denomina- 
tions, and  their  sum  will  be  the  answer  required. 

Note. — If  there  are  no  feet  in  the  multiplier,  supply  their 
place  with  a  cipher,  and  in  like  manner  supply  the  place  of  any 
other  denomination  between  the  highest  and  lowest. 


Feet  X  by  feet  give  feet. 
Feet  X  by  inches  give  inches. 
Feet  X  by  seconds  give  sec'ds. 


Inches  xby  inches  give  seconds. 
Inches  X  by  seconds  give  thirds. 
Seconds  xby  sec.  give  fourths. 


Note. — Let  it  be  remembered,  that  though  the  feet  obtained 
by  the  rule  are  square  feet,  the  inches  are  not  square  inches, 
but  the  twelfth  part  of  a  square  foot,  ^^, 

8.  Multiply  2-i  feet  by  2i  feet. 

Thus:     2— 6in.     or,  2i     or,  2.5  decimally. 


2     6  2i  2.5 


5     0  5-0       125 

1     3     0         1—}       50 


Ans.  feet    6-3  ft.  6^  6.25 


So  that  3  is  not  3  inches,  but  36  inches,  or  |^  or  a  square  foot 
t\V  12x3=tYt,  &c. 


DUODECIMALS.  $83 

9.  Multiply  10  feet  6  inches  by  4  feet  6  inches  ? 
Thus  :         10—6  in.     or,  10-6     or,  10.5 
4—6  in.  4-6         4.5 


5     3     0          42       0 
42     0                  5       3 

525 

420 

ft.  47—3     0  Ans.  47 3 

As  before,  47i  feet.=^/y. 

10.  Multiply  9  ft.  8',  6''  by  7  ft.  9',  3'\ 

ft.      '       " 

9       8       6 

7       9       3 

47.25 

67 — 11 6=product  of  the  feet  in  the  multiplier. 

7       3       4     ^'"         do.  the  primes. 

2       5     1  ^''"    do.  by  the  seconds. 

ft.  75 5 3—7-6  Ans, 

11.  Multiply  9  ft.  7  in.  by  3  ft.  6  in.         Ans,  33  ft.  6  in.  6'. 

12.  Multiply  3  ft.  11  in.  by  9  ft.  5  in.  36  ft.  10  in.  7', 

13.  Multiply  8  ft.  6  in.  9^'  by  7  ft.  3  in.  8'.    62  ft.  6  in.  7",  9"'. 

14.  In  a  load  of  wood,  8  feet  4  Thus  :  8  ft.  4  in.  length, 
inches  long,  2  feet  6  inches  high,  2  6  X  height, 
and  3  feet  3  inches  wide,    how  

many  solid  feet?  16       8 

4       2+ 


20     10  sum. 
3       3  X  width. 


62       6 
5       2     6'^--f- 

Cub.  p.  od.  67       8     ^" ,  Ans. 

15.  How  many  solid  feet  are  there  in  a  stick  of  timber,  70 
feet  long,  15  inches  thick,  and  18  inches  wide  ?  Thus  :  70  ft. 
Xl5  in.  =  1050xl8  in.  =  18900''~144  =  131,  and  36''  rem., 
fhen  36-^12  =  3  in.  Ans.  131i  ft. 

16.  How  many  cubic  feet  of  wood  in  a  load  6  ft.  7  in.  long, 
3  ft.  5  in.  high,  and  3  ft.  8  in.  wide  ?  Ans.  82  ft.  5'  8''  4''". 

When  the  feet  of  the  multiplier  exceed  twelve. 
Rule. — Multiply  by  the  feet  of  the  multiplier  as  in  compound 
multiplication,  and  take  parts  for  the  inches   &c. 

20* 


284  DUODECIMALS. 

17.  Multiply  112  feet,  3  inches,  5''  by  42  feet,  4  in.,  6''  ? 
112  ft.         3  in.      5'' 

6x7=42 


673 

8 

6 
7x 

4715 
4  in.  1  =  37 
6''       i=    4 

11 
5 

8 

6 

1     8''' 

1     8     6"'^ 

Ans.  4758  0  9     4     6''^' 

18.  What  is  the  freight  of  a  bale  of  goods  containing  65  feet 
9  inches,  at  D15  per  ton  of  40  feet? 

20  ft.     1=15.00  Or,  65.75 

5  ft.     i=  7.50  15  X 

6  in.yV=    1-87.5  

2  in.    1=      18.7  40)986.25 

9.3  

D24.65.6  Ans. 


D24.65.5 


19.  What  will  be  the  expense  of  plastering  the  wall  of  a  room 
8^  feet  high,  and  each  side  16  feet  3  inches  long,  at  50  cents 
per  square  yard  ?  Ans.  D30.69.4. 

20.  How  many  square  feet  in  a  board  17  feet  7  inches  long, 
1  foot  5  inches  wide  ?  Ans.  24  ft.  10^^  11''. 
,  21.  A  load  of  wood  is  4  feet  6  inches  wide,  3  feet  10  inches 
high,  and  7  feet  8  inches  long ;  how  many  feet  more  than  a  cord 
does  it  contain  1  Ans.  4^  feet. 

22.  What  will  the  paving  of  a  courtyard  cost,  which  is  70 
feet  long,  and  56  feet  4  inches  wide,  at  20c.  per  square  foot  ? 

Ans.  D788  66| 

23.  A  man  built  a  house  consisting  of  3  stories  ;  in  the  upper 
story  there  were  10  windows,  each  containing  12  panes  of 
glass,  each  pane  14  by  12  inches  ;  the  first  and  second  stories 
contained  14  windows,  each  15  panes,  and  each  pane  16  by  12; 
how  many  square  feet  of  glass  were  there  in  the  whole  house  ? 

Ans.  700  sq.  ft 

REVIEW. 

What  are  Duodecimals  ?  By  whom,  and  how  can  those  rules 
be  applied  ?  What  is  the  rule  for  addition  ?  subtraction  ?  mul- 
tiplication ?  &c 


235 

APPENDIX, 

Containing  a  variety  of  useful  rules  and  examples  in  square  and 
cubical  measure,  mensuration,  <Sfc. 

ARTIFICERS'  WORK. 

Artificers  estimate  or  compute  the  value  of  their  work  by 
different  measures  ;  but  the  best  method  of  taking  the  dimen- 
sions of  all  sorts  of  work  is  by  feet,  tenths,  and  hundredths  ;  in 
other  words,  by  decimals  ;  glazing,  and  masons'  flat  work,  &c., 
by  the  foot ;  painting,  plastering,  paving,  by  the  yard  ;  flooring, 
partitioning,  roofing,  tiling,  by  the  square  of  100  feet;  brick- 
work, by  the  rod  of  16-^  feet,  whose  square  is  272.25. 

Note. — In  calculating  the  square  feet  it  is  usual  to  omit  the  25, 
but  the  more  correct  way  is  to  use  the  perfect  number  (272.25). 

The  practice  has  formerly  been  to  calculate  the  various  prices 
of  mechanical  work,  according  to  the  rules  and  regulations  of  the 
several  countries  of  Europe,  particularly  England  and  Germany, 
but  the  method  has,  in  some  respects,  been  departed  from, 
which  has  rendered  the  calculations  different  in  the  several 
states  of  the  Union.  The  general  system  of  computation  will 
DC  given,  from  which  any  others  may  easily  be  derived. 


BRICKLAYERS'  WORK. 

Bricklayers  compute  their  work  at  the  rate  of  one  brick  ana 
a  half  thick,  and  if  the  wall  be  more  or  less  than  this  standard, 
it  must  be  reduced  to  it  as  follows  : — 


Multiply  the  superficial  content  of  the  wall,  in  feet,  by  the 
number  of  half  bricks  in  thickness,  and  ^  of  that  product  will 
be  the  content  required. 

It  is  generally  the  practice  in  this  country,  at  the  present 
time,  to  calculate  bricks  by  the  1000.  All  windows,  doors,  &c., 
are  to  be  deducted  out  of  the  content  of  the  walls  in  which 
they  are  placed  ;  but  this  deduction  is  made  only  with  regard  to 
materials  ;  for  the  value  of  their  workmanship  is  added  to  the 
bill  at  the  rate  agreed  on. 

1.  How  many  square  rods  are  there  in  a  wall  52.5  feet  long, 
12.75  feet  high,  and  2.5  bricks  thick  ?  Thus  :  52.5  X  12.75-t- 
272=2.4609x5  half  bricks  =  12.3045-^3=4.1015=4  rodS;  27 
feet,  7  inches.  Ans. 


236  masons'  work. 

2.  How  many  square  rods  are  there  in  a  wall  62^  feet  k)ng, 
14  feet  8  inches  high,  and  2^  bricks  thick  1 

Ans.  5  rods,  167  feet,  9  inches,  4  pa. 

3.  How  many  bricks,  8  inches  long,  4  inches  wide,  2^  inches 
thick,  will  build  a  wall  in  front  of  a  garden,  which  is  to  be  240 
feet  long,  6  feet  high,  1  foot  6  inches  wide  ?  Ans.  51840  bricks. 

4.  How  many  bricks,  9  inches  long,  5  inches  wide,  2^  thick, 
will  it  require  to  build  a  wall  90  feet  long,  7  feet  high,  and  2 
feet  thick ;  and  what  will  it  cost  at  90  cents  per  square  foot  1 


MASONS'  WOilK. 

To  masonry  belong  all  sorts  of  stone-work  ;  and  the  measure 
made  use  of  is  a  solid  perch,  or  a  superficial,  or  a  solid  foot. 
Solid  measure  is  generally  used  for  materials,  and  the  superficial 
for  workmanship. 

RULE. 

In  solid  measure  multiply  the  length,  breadth,  and  thickness, 
continually  together,  and  in  superficial  measure  the  length  and 
breadth  of  every  part  of  the  projection  must  be  taken. 

1.  Required  the  solid  content  of  a  wall,  whose  length  is  48.5 
feet,  its  height  10.75  feet,  and  thickness  2  feet. 

48.5X10.75X2  =  1042.75  feet.  Ans, 

2.  What  is  the  solid  content  of  a  wall  whose  length  is  60.75 
feet,  its  height,  10.25  feet,  and  its  thickness  2^  feet? 

Ans,  1556.71875  feet. 

3.  What  is  a  marble  slab  worth,  whose  length  is  5  feet  7 
inches,  and  breadth  1  foot  10  inches,  at  80  cents  per  foot  ? 
Thus  :  5  ft.  7  in.=5/2  ft.=:f|  and  1  ft.  10  in.=lf  ft.  =  U  ft. 
{\  X  V  =  W  ^content  of  the  slab  in  feet,  \y  x  80c.  =  ^\^2^0c. 
=  7  3_7  0c.^D8.18.8|.  Ans. 

4.  How  many  solid  perches  of  stone  are  contained  in  a  cellar 
wall,  the  length  being  45.5  ft.  on  a  side,  and  the  breadth  24  ft.  at 
each  end,  6.75  feet  high,  and  2  feet  thick  ?  Ans.  1 13.72fperches. 

5.  Required  the  cost  of  making  a  stone  wall  under  a  build- 
ing, whose  length  is  42  feet,  breadth  on  the  outside  26  feet,  the 
height  of  the  wall  being  6.5  feet,  and  2  feet  thick,  at  40  cents 
per  solid  perch.  Ans.  D40.33.6  + 

6.  In  a  block  of  marble  5  ft.  long,  18  in.  square,  how  many 
cubical,  and  how  many  square  feet?  18  in.  X  18x60  in.= 
19440-1728=111  cub.  ft.  ;  19440-r  144  =  135  sq.  ft.  Ans. 


SLATERS    AND    TILERs'    WORK.  237 

CARPENTERS  AND  JOINERS'  WORK. 

Carpenters  and  joiners'  work  is  that  of  flooring,  partitioning, 
roofing,  &c.,  and  is  measured  by  the  square  of  100  feet. 

1.  If  a  floor  be  57.25  feet  long,  and  28.5  feet  broad,  how 
many  squares  will  it  contain?  Thus,  57.25x28.5  =  1631.625 
square  feet,  =:16  squares,  31  feet,  7  inches,  Q" .  Ans. 

2.  A  partition  is  91.75  feet  long,  and  11.25  feet  broad  ;  how 
many  squares  does  it  contain?  Ans,  10  squares,  32  feet. 

3.  A  partition  is  96.75  feet  long,  11 .5  feet  broad ;  how  many 
squares  will  it  contain?  Ans.  11.12625  squares. 

4.  What  is  the  expense  of  flooring  a  building  45.5  feet  long 
26.75  wide, 2  stories  high, at  Dl. 36  per  square?  Ans.  D49.65.87. 

5.  If  a  floor  be  60  feet  long,  28.75  broad,  how  many  squares 
will  it  contain?  Ans.  17.25  squares. 

6.  In  a  floor  46  by  24  feet,  required  the  expense  of  flooring 
at  15  cents  per  square  foot,  and  cost  of  boards  at  D7.50  per  M. 

Ans.  Cost  of  flooring,  D16.56  ;   cost  of  boards  D8.28. 


SLATERS  AND  TILERS'  WORK. 

In  these  works,  the  content  of  a  roof  is  found  by  multiplying 
the  length  of  a  side  by  the  girth  from  eave  to  eave  ;  and  in  sla- 
ting, allowance  must  be  made  for  the  double  row  at  the  bottom. 
In  taking  the  girth,  the  line  is  made  to  ply  over  the  lowest  row 
of  slates,  and  returned  up  the  under  side  till  it  meet  with  the 
wall  or  eaves-board ;  but  in  tiling,  the  line  is  stretched  down 
only  to  the  lowest  part,  without  returning  it  up  again.  Double 
measure  is  generally  allowed  for  hips,  valleys,  gutters,  &c.,  but 
no  deductions  are  made  for  chimneys.  In  all  works  of  this 
kind,  the  content  is  computed  either  in  yards  of  9  square  feet, 
or  in  squares  of  100  feet,  and  the  same  allowance  of  hips  and 
valleys  is  to  be  made  as  in  roofing. 

1.  The  length  of  a  slated  roof  is  45.75  feet,  and  its  girth 
34.25  feet;  required  the  content.  Thus:  45.75x34.25  = 
1566.9375  square  feet=  174.104  yards.  Ans. 

2.  The  length  of  a  slated  roof  is  48.5  feet,  and  its  girth 
36.25  feet;  required  the  content.      Ans.  1758.125  square  feet. 

3.  What  will  the  tiling  of  a  barn  cost  at  D3.40  per  square, 
he  length  being  43  feet  10  inches,  and  the  breadth  27  feet  5 

inches  on  the  flat,  the  eave-board  projecting  16  inches  on  each 


238  PAINTERS    AND    GLAZERS'    WORK. 

side,  allowing  the  roof  to  be  a  true  pitch  1     Thus,  27  feet  5 

inrliPQ  —  975    329       ,,^^1     32J    I    /32_?_^9\ — 329il64.5 493.5 

incnes_.i/y^_  j2^ ,  ana  y2  '  V  T2.  ~'^i— T2  +  T2  —  T2 
=  true  pitch.  16  x  2  =  32  inches,  |^  feet,  to  be  added  for  pro- 
jection ;  then,  4||-^  +  ff=^^|^=girt  of  roof;  again,  43  feet 
10  inches=43f  feetrrr2G3^  and  ^ 25.5  ^  2|3_i3  8|o 6.5  fg^^^ 
138|^65  squares  of  roofing;  hence  1382065  x  D3.40~ 
469|02.i  cents  =:D65.26.4m.,  cost  of  roofing.  Ans. 


PLASTERERS'  WORK. 

Plasterers'  work  is  of  two  kinds,  namely,  plastering  upon 
laths,  called  ceiling ;  and  plastering  walls,  called  rendering ; 
and  these  different  kinds  must  be  measured  separately,  and  their 
content  collected  into  one  sum.  Their  work  is  measured  by 
the  square  foot,  or  yard  of  9  square  feet,  and  moulding  by  run- 
ning measure. 

1.  If  a  ceiling  be  59.75  feet  long,  and  24.5  feet  broad,  how 
many  yards  does  it  contain?  Thus:  59.75x24.5  =  1463.875 
square  feet=  162.6528  square  yards.   Ans. 

2.  If  the  partitions  between  rooms  are  141.5  feet  about,  and 
11.25  feet  high,  how  many  yards  do  they  contain? 

Ans.  176.87. 

3.  If  a  ceiling  be  64.75  feet  long,  and  24.5  broad,  how  many 
square  yards  does  it  contain?  Ans.  176.263|. 


PAINTERS  AND  GLAZERS'  WORK. 

Painters'  work  is  measured  in  the  same  manner  as  that  of 
carpenters  ;  and  in  taking  the  dimensions,  the  line  must  be  forced 
into  all  the  mouldings  and  corners.  The  work  is  estimated  by 
the  yard,  except  sashes,  which  are  calculated  per  light.  Gla- 
zers'  work  is  calculated  by  the  light.  All  work  of  this  kind  is 
done  by  the  square  yard  of  9  feet. 

1.  If  a  room  be  painted,  whose  height  is  16.5  feet,  and  its 
compass  97.75  feet,  how  many  yards  does  it  contain  ? 

Thus:  97.75X16.5  =  1612.875  square  feet=179.208  square 
yards.  Ans, 


pavers'  work.  239 

2.  The  height  of  a  room  is  14  feet  10  inches,  and  the  cir- 
cumference 21  feet  8  inches  ;  how  many  square  yards  does  it 
contain?     Thus:     14—10 
21 8x 


9—10-8 
311 6 

321       4  8 
Then,  321.3888  feet  =35.71  square  yards  +.  Ans. 


PAVERS'  WORK. 

Pavers'  work  is  done  by  the  square  yard.  The  content  is 
found  by  multiplying  the  length  by  the  breadth. 

1.  How  many  square  yards  in  a  rectangular  (right-angled) 
court-yard,  the  length  being  27  feet  10  inches,  and  breadth  14 
feet  9  inches  ?    Ans.  410  ft.  6  in.  6  pa.=:45  yds.  7  ft.  4  in.  8—8. 

2.  A  rectangular  court-yard  is  64.75  feet  long,  and  45.5  in 
breadth ;  what  is  the  expense  of  paving  at  45  cents  per  square 
yard?  Ans.  D  147.30. 

The  dimensions  of  the  walls  of  a  brick  building  being  given,  to 
find  how  many  bricks  are  required  to  build  it* 


From  the  whole  circumference  of  the  wall  measured  round 
on  the  outside,  subtract  4  times  its  thickness  ;  then  multiply  the 
remainder  by  the  height,  and  that  product  by  the  thickness  of 
the  wall,  will  give  the  solid  content  of  the  whole  wall,  which 
multiplied  by  the  number  of  bricks  contained  in  a  solid  foot, 
will  give  the  answer. 

1.  How  many  bricks  8  inches  long,  4  inches  wide,  2.5  inches 
thick,  will  it  require  to  build  a  house  44  feet  long,  40  feet  wide, 
and  20  feet  in  height,  the  wall  to  be  1  foot  in  thickness  ;  thus, 
8x4x2.5=80  solid  inches  brick  ,  1728-^80=21.6  bricks  in  a 
solid  foot;  44  +  40  +  44-|-40=168  feet  length  of  wall;  168  — 
4=164x20x21.6=1 70848.0  bricks.  Ans. 

2.  In  a  room  of  4  walls,  2  of  them  measure  12.5  feet  in  length, 
and  7.5  in  height,  and  the  other  two  sides  are  14.6  feet  in  length, 


240  MENSURATION. 

by  7.5  feet  in  height ;  required  the  number  of  square  yards  ;  and 
Avhat  the  plastering  will  amount  to  at  I2c.  per  yard.  Thus: 
12.5x7.5x2  =  187.5  feet;  14.5  X  7.5  X  2=217.5  feet ;  217.5-f- 
187.5=405.0~9=:45  yds.  Xl2=D5.40.  Ans. 

3.  How  many  square  yards  in  a  garden  of  96  by  50  feet  ? 

Ans.  533^  yards. 

To  calculate  the  number  of  shingles  for  a  roof 

I 

RULE. 

1.  Reduce  the  length  and  breadth  of  the  space  to  be  roofed  to 
inches  separately. 

2.  Divide  the  breadth  by  the  average  width  of  the  shingles, 
and  the  quotient  will  be  the  number  of  shingles  in  one  course. 

3.  Divide  the  length  by  the  number  of  inches  you  intend  lay- 
ing the  shingles  or  courses  of  the  weather,  and  the  quotient  will 
be  the  number  of  courses. 

4.  Multiply  the  number  of  courses  by  the  number  of  shingles 
in  one  course,  and  you  will  have  the  number  of  shingles  re 
quired 

1.  Required  the  number  of  shingles  for  a  roof  16  feet  in 
width  and  18  feet  in  length,  the  shingles  to  average  5  inches 
each  in  width,  and  the  course  to  run  8  inches  to  the  weather. 

Thus:   16x12  =  192-^5  =  38.4;    18x  12=216-~8:=27x 
38.4  =  1036.8  shingles.  Ans. 


MENSURATION,  &c. 
To  measure  wood,  (^c. 

RULE. 

Multiply  the  width  by  the  height,  that  product  by  the  length, 
and  divide  by  128,  and  the  quotient  is  the  answer  in  cords. 

1.  Required  the  content,  in  cords,  of  a  pile  of  wood  16  feet 
in  length  5.5  feet  in  width,  and  4.5  feet  in  height.  4.5x5.5  X 
16  =  396.00-^128  =  3  cords.  Ans. 

2.  How  many  cubical  feet  in  a  piece  of  scantling  40  feet  in 
length,  1 .5  feet  in  width,  and  .5  feet  in  depth  ?  1.5X  .5  X  40= 
30.00  feet.  Ans. 


Mensuration.  241 

3.  In  a  tier  of  wood  25  feet  in  length,  18.5  feet  in  width,  and 
7.5  feet  in  height,  how  many  cords  ?       Ans.  27  cords,  12  feet. 

4.  Required  the  cost  of  a  load  of  wood  9  feet  in  length,  4.5 
feet  in  height,  and  4  feet  3  inches  in  width,  at  D5,50  per  cord  ? 

Thus  :  4.50  X  4.25  X  9=  172.125  ;  then  128  :  5.50  ;:  172.125  : 
D7.39.6.  Ans. 

5.  How  many  solid  feet  of  timber  in  a  stick  8  feet  long,  10 
inches  thick,  and  6  inches  in  width  1  Ans.  3J 

In  10  feet  long,  12  inches  thick,  and  1  foot  3  inches  wide  ? 

Ans.  121 

6.  In  a  pile  of  wood  10  feet  wide,  3  feet  3  inches  high,  and 
1  mile  long,  how  many  cord  feet,  and  how  many  cords  ?  10725 
cord  feet==1340|-  cords. 

7.  How  many  cubical  feet  in  a  pile  of  rails  70  feet  long,  cut 
12 J  feet  and  14^  feet  high  ;  and  how  many  cords  of  wood  would 
they  make?  ^/^/|    £>viL^f 

To  find  the  area  of  a  square  having  equal  sides,    i 


Multiply  the  side  of  the  square  into  itself,  and  the  product 
will  be  the  area,  or  content. 

1.  In  a  garden  130  feet  square,  how  many  square  yards  ? 

Ans.  1600. 

To  measure  a  rectangle  parallelogram  ^  or  long  square. 

A  rectangle  is  a  four-sided  figure  like  a  square,  in  which  the 
sides  are  perpendicular  to  each  other  (right-angled),  but  the  ad' 
jacent  sides  are  longer  and  parallel. 


RULE. 


Multiply  the  length  by  the  breadth,  and  the  product  will  be 
the  answer. 

1.  A  garden  is  76  feet  in  length,  and  42  feet  in  width ;  how 
many  square  feet  of  ground  are  contained  in  it  ? 

Ans.  76X42=3192  feet. 

2.  What  is  the  content  of  a  field  40  rods  square  ? 

Ans.  10  acres. 

3.  What  is  the  content  of  a  field  25  chains  long  by  20  chains 
broad?  Ans.  50  acres, 

4.  Required  the  content  of  a  field  75  chains  long  and  75 
chain*  broad  ?  rr&rn^  -     i  C  ti^ 

21 


242 


MENSURATlvJN. 


Note. — In  measuring  boards,  you  can  multiply  the  length  ir> 
feet  by  the  breadth  in  inches,  and  divide  by  12  ;  the  quotien 
willgive  the  answer  in  square  feet. 

5.  In  a  board  20  feet  in  length,  16  inches  in  width,  how 
many  feet?  ^;z5.  20x  16=320-M2 


26f  feet. 


To  measure  a  triangle,  or  to  find  the  area. 


A  triangle  is  a  figure  bound- 
ed by  three  straight  lines  ; 
thus,  B,  A,  C,  is  a  triangle. 
When  a  line  like  A,  D,  is 
drawn,  making  the  angle  A 
D  B  square  to  the  angle  A 
D  C,  then  A  D  is  said 
to  be  perpendicular  to  B  C, 
and  A  D  is  called  the  altitude 
of  the  triangle.  Each  triangle, 
B  A  D,  or  D  A  C,  is  called 
a  right-angled  triangle.  The 
side  B  A,  or  the  side  A  C, 
opposite  the  right  angle,  is 
called  the  hypotenuse. 


Multiply  the  base  of  the  given  triangle  into  half  its  perpen* 
dicular  height,  or  half  the  base  into  the  whole  perpendicular, 
and  the  product  is  the  answer. 

1.  Required  the  area  of  a  triangle  whose  base  or  longest  side 
is  32  inches,  and  the  perpendicular  height  14  inches. 

Arts. '^2x1  (^  of  14)  =  224  square  inches. 

2.  In  a  triangular  field  the  base  is  40  chains,  and  the  perpen- 
dicular 15  chains  ;  how  many  acres  ? 

Ans.  15x40  =  600-^2=300—10  ch.  =30  acres. 

3.  What  is  the  area  of  a  square  piece  of  land,  of  which  the 
sides  are  27  chains  ?  .  Ans.  72  acres,  3  roods,  24  poles. 

4.  How  many  acres  in  a  piece  of  land  560  rods  long  and  32 
rods  wide  ?  Ans.  112  aq|ps, 

5.  How  many  acres  are  contained  in  a  road  40  miles  long 
and  4  rods  wide  ?  Ans.  320. 

6.  What  will  a  lot  of  land  1  mile  square  come  to,  at  D20.75 
per  acre?  Ans.  D 1 3280. 

7.  How  many  yards  of  carpeting,  that  is  1^-  yards  wide,  will 
i;over  a  floor  21  feet,  3  inches  long,  and  13  feet,  6  inches  wide  ? 

Am>  ^5J  yarcl^ 


MENSURATIO.V.  243 

To  measure  a  circle,  area,  circumference,  d^c. 

A  circle  is  a  portion  of  a  plane 
bounded  by  a  curved  line,  every  part 
of  which  is  equally  distant  from  a 
certain  point  within,  called  the  cen- 
tre. The  curved  line,  A  E  B  D,  is 
called  the  circumference  ;  the  point 
C  the  centre  ;  the  line  A  B,  pas- 
sing through  the  centre,  a  diameter, 
and  C  H  the  radius.  The  circum- 
ference A  E  B  D  is  3.1416+times 
greater  than  the  diameter  A  B. 
Hence,  if  the  diameter  is  1,  the  circumference  will  be  3. 1416 -f- 
Also,  if  the  diameter  is  known,  the  circumference  is  found  by 
multiplying  3.1416  by  the  diameter.  Hence  the  following  rule: — 

RULE. 

Multiply  the  diameter  by  3.1416,  and  the  product  Avill  be  the 
circumference  ;  or  divide  the  circumference  by  3.1416,  and  the 
quotient  will  be  the  diameter. 

Note  1. — As  7  is  to  22,  so  is  the  diameter  to  the  circumfer- 
ence ;  or,  as  22  is  to  7,  so  is  the  circumference  to  the  diameter. 
Or,  more  correctly: — 

As  113  is  to  355,  so  is  the  diameter  to  the  circumference  ; 
or  as  355  is  to  113,  so  is  the  circumference  to  the  diameter. 

Note  2. — The  problem  of  "  squaring  the  circle,"  as  it  is 
usually  termed,  has  never  been  solved,  nor  can  a  square  or  any 
other  right-lined  figure,  be  found,  that  shall  be  equal  to  a  given 
circle,  as  there  rnust  be  a  fraction  in  one  case  or  the  other ;  it  is 
not  in  the  power  of  numbers  to  bring  them  exactly  alike  ;  the 
numbers  used  above  are  sufficiently  correct ;  but  the  calculation 
maybe  extended  to  almost  an  indefinite  number  of  decimals,  with- 
out apparently  arriving  any  nearer  the  solution  of  the  problem. 

1.  If  the  circumference  of  a  circle  be  354,  what  is  the  diam- 
eter ?     Thus:  354.000-^3.1416=112.681,  diameter.  Ans. 

2.  If  the  diameter  of  a  circle  be  17,  what  is  the  circumfer- 
en<;e  ?     Thus  :  3.1416  x  17  =  53.4072,  circumference.  Ans. 

3.  If  the  circumference  of  the  earth  be  25000  miles,  what  is 
the  diameter?  Ans.  7958  miles  (nearly). 

4.  The  base  of  a  cone  is  a  circle  ;  what  is  its  diameter,  when 
the  circumference  is  64  feet?  Ans.  20.3718  '^ 

5.  What  is  the  circumference  of  a  wheel  whose  diameter  is 
5  feet,  2  inches  ?  Ans.  16.2316. 

6.  If  the  circumference  of  a  carriage- wheel  be  16  feet,  6 
inches,  what  is  the  diameter  ?  Ans.  5  252Weet,'f 


2^4  MSNSURATtOIf. 

7.  The  circumference  of  a  circle  is  16  chains  ;  what  is  tha 
diameter?  ^W5.  5.0929+chains, 

The  diameter  given^  to  find  the  area  or  content. 

RULE. 

Multiply  the  square  of  the  diameter  by  the  decimal  .7854, 
and  the  product  will  be  the  area. 

1.  How  many  square  feet  are  contained  in  a  circle,  whose 
diameter  is  4  feet  3  inches?  Thus:  4.25^  =  180625  X  .7854  = 
14.1862875  square  feet.     Ans. 

2.  What  is  the  value  of  a  circular  garden,  whose  diameter  is 
6  rods,  at  the  rate  of  8  cents  per  square  foot  ? 

Ans.  D615.8L6.432. 
The  area  af  a  circle  given^  to  find  the  diameter, 

RULE. 

Divide  the  area  by  .7854,  and  extract  the  square  root  of  the 
quotient. 

1.  The  area  of  a  circle  is  5  acres,  3  roods,  26  perches,  re- 
quired the  diameter.  Thus  :  946.000-^-.7854=34. 7  poles.  Ans. 

2.  What  is  the  length  of  a  rope  fastened  to  a  stake  in  the 
centre  of  a  circular  field,  and  the  other  end  to  the  nose  of  a 
horse,  which  will  permit  him  to  feed  on  2  acres  of  land  ? 

Ans,  2.5231  chains. 

The  circumference  given^  to  find  the  area, 

RULE. 

Multiply  the  square  of  the  circumference  by  the  decimal 
.07958,  and  the  product  will  be  the  area. 

If  the  circumference  of  a  circle  be  1,  the  diameter  =1  — 
3.14159=0.31831  ;  and  \  the  product  of  this  into  the  circum- 
ference is  .07958,  the  area. 

1.  If  the  circumference  of  a  circle  be  136  feet,  what  is  the 
area  1  Ans.  1472  feet. 

2.  The  circumference  of  a  circle  being  37.7,  required  the 
area.  Thus:  37.72  ==1421.29=square,X. 07958  =  113.1062582 
=  area  of  the  circle. 

To  find  the  surface  of  a  sphere,  globe ^  or  hall. 

RULE. 

Square  the  diameter,  and  multiply  it  by  the  decimal  3.1416, 
~  and  the  product  will  be  the  answer. 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  12  ? 

Ans.  122-144x3.1416=452.3904 

2.  Required  the  number  of  square  inches  in  the  surface  of  a 
sphere  whose  diameter  is  2  feet,  or  24  inches  ?  Ans.  1809.5616. 


MENSURATION.  245 

To  find  the  solidity  of  a  sphere,  ^^^^ 

A  sphere  or  globe  is  a  round  solid  body.  ^^to-.^^k 

Multiply  the  surface  by  the  diameter  and  divide  the      sphere. 
product  by  6  ;  the  quotient  will  be  the  solidity   (multiply  the 
square  of  the  diameter  by  3.1416). 

1.  What  is  the  solidity  of  a  sphere  whose  diameter  is  12  ? 
Thus:  122=144x3.1416  =  452.3904x12  =  5428.6848-6= 

904.7808,  Arts.       Or,  multiply  the  cube  of  the  diameter  bv 
.5236,  thus:    12  X  12x12  X  .5236  =  904.7808. 

2.  What  is  the  solidity  of  the  earth,  its  mean  diameter  being 
7918.7  miles  ?  Ans,  259992792082.6374908. 

Tofi^nd  the  solidity  of  a  prism. 
Definition. — A  prism  is  a  body  with  two  equal     ^^^ 
or  parallel  ends,  either  square,  triangular,  or  poly-    '"'"^ 
gonal,  and  three  or  more  sides,  which  meet  in         Prism, 
parallel  lines,  running  from  the  several  angles  at  one  end  to 
those  of  the  other. 

RULE. 

Multiply  the  area  of  the  base  by  the  altitude,  and  the  product 
will  be  the  content. 

1.  What  is  the  content  of  a  prism,  each  side  of  the  square 
which  forms  the  base  being  15,  and  the  altitude  of  the  prism  20 
feet?  Ans.  152=225x20=4500  feet. 

2.  The  side  of  a  stick  of  timber  is  hewn  3  square,  is  10 
inches,  and  the  length  is  12  feet;  required  the  content. 

Ans.  10x4.33=1  perp. =43. 3  area  at  the  end,  X  12  feet 
length,  =519.6^144  =  3.6+  content. 

3.  Required  the  solidity  of  a  triangular  prism  whose  height 
is  10  feet,  and  area  of  the  base  350  feet,  Ans.  3500  feet 

To  find  the  convex  surface  of  a  cylinder. 
Definition. — A  cylinder  is  a  round  body,  whose 
bases  are  circles,  like  a  round  column,  or  stick  of  ^  ________ 

timber,  of  equal  bigness  at  both  ends.  Cylinder. 

RULE. 

Multiply  the  circumference  of  its  base  by  the  altitude. 

1.  What  is  the  convex  surface  of  a  cylinder,  the  diameter  of 
whose  base  is  20,  and  the  altitude  50  feet  ? 

Ans.  3.1416x20x50  =  3141.6000- 

2.  Required  the  convex  surface  of  a  cylinder,  the  circumfer 
•nr    c^r  whosft  base  is  6.509.  and  altitude  27.  Ans,  175743 


246  MENSURATION. 

To  find  the  solidity  of  a  cylinder. 


Multiply  the  square  of  the  diameter  of  the  end  by  .7854, 
which  will  give  the  area  of  the  base  ;  then  multiply  the  area  of 
the  base  by  the  length,  and  the  product  will  be  the  content. 

1.  Required  the  solid  content  of  a  round  stick  of  timber  of 
equal  bisrness  at  both  ends,  whose  diameter  is  1.5  and  length  20 
feet.     Thus:   1.5  X  1.5=2.25  X. 7854  X  20=35.3430.  Ans. 

Or,  18X  18  =  324  X  .7854  X  20=5089.3920-r- 144  =  35.3430. 
Ans. 

2.  What  is  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  12,  and  the  altitude  30  ?  Ans,  3392.928. 

3.  How  many  solid  feet  in  a  round  stick  of  timber  16  feet 
long,  and  the  diameter  at  each  end  15  inches? 

Ans.  19.635  solid  feet. 

4.  Required  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  30  inches,  and  the  height  50  inches  ? 

Ans.  20.4531  solid  feet 

To  find  the  solidity  of  a  cone. 
Definition. — A  cone  is  a  round  solid  body  of  a  true  ta- 
per from  the  base  to  a  point,  which  is  called  the  vertex. 


RULE. 


Cone. 


Multiply  the  area  of  the  base  by  the  altitude,  and  divide 
the  product  by  3  ;  that  is,  square  the  diameter  and  multiply  it 
by  .7854,  which  gives  the  area  of  the  base  ;  then  multiply  by 
the  altitude  and  -f-by  3.  Or,  the  square  of  the  circumference 
of  the  base  X  by  .07958,  and  that  product  by  ^  of  the  perpcn 
dicular  altitude,  and  the  product  will  be  the  solidity. 

1.  Required  the  solidity  of  a  cone,  the  diameter  of  who^e 
base  is  5  and  the  altitude  10. 

Ans.  52=25  X. 7854=  19.635  XlO-r  3  =  65.45 

2.  What  is  the  solidity  of  a  cone  whose  altitude  is  27  feet 
and  the  diameter  of  the  base  10  feet  ?  Ans.  706.86 

3.  Required  the  solidity  of  a  cone,  the  diameter  of  whoso 
base  is  18  inches,  and  its  altitude  15  feet.     Ans.  8.83575  feet. 

4.  If  the  circumference  of  the  base  of  a  cone  be  40  feet,  and 
the  height  50  feet,  what  is  the  solidity?     Ans.  2122.1333  feet 

5.  The  top  of  a  cistern  is  5.5  feet,  the  bottom  4.75  feet,  and 
the  height,  or  depth,  7.25 ;  how  many  hogsheads  will  it  con 
tain?  Ans.  17.7'=i-f- 


MENSURATION.  24T 

To  find  the  solidity  of  a  pyramid. 
Definition. — A  pyramid  is  a  solid,  whose  sides  are  all 
triangles,  meeting  in  a  point  at  the  vertex,  and  the  base 
any  plane  figure  whatever. 

Pyramid. 
RULE. 

Multiply  the  area  of  the  base  by  the  altitude,  and  divide  the 
product  by  3. 

1 .  Required  the  solidity  of  a  pyramid,  of  which  the  area  of  the 
base  is  95  and  the  altitude  15.      Ans.  95  X  15=1425-^3=475. 

2.  What  is  the  solidity  of  a  pyramid,  the  area  of  whose  base 
is  403  and  the  altitude  30  ?  Ans,  4030. 

3.  Required   the    solidity   of  a   triangular   pyramid,   whose 
height  is  30,  and  each  side  of  the  base  3  ? 

Thus:  32  X. 433013  =  3. 897117  area  of  the  base,  then  3.897117 
X  33°n:38.97117=solidity  required. 

4.  A  pyramid  with  a  square  base,  of  which  each  side  is  30, 
has  an  altitude  of  20  ;  what  is  the  solid  content  ?       Ans.  6000. 

To  measure  a  parallelopipedon. 
Definition. — A  parallelopipedon  is   a  solid   of 
three  dimensions,  length,  breadth,  and  thickness  ; 


as  a  piece  of  timber  exactly  square,  whose  length  paraiiei^ipedon. 
is  more  than  the  breadth  and  thickness  ;  ^he  ends  are  called 
bases,  which  are  equal.  The  solidity  of  a  parallelopiped  is 
equal  to  the  product  of  the  base  into  the  perpendicular  altitude. 
And  a  parallelopiped  and  a  cylinder,  which  have  equal  bases 
and  altitudes,  are  equal  to  each  other. 

RULE. 

Find  the  area  of  the  base,  then  multiply  that  by  the  length, 
and  it  will  give  the^solid  content. 

1.  If  the  side  of  a  stick  of  timber  is  1.75  feet,  and  the  length 
9.5  feet,  to  find  the  content.  Thus:  1.752=3.0625==  area  of 
base  x9.5  =29.09375  content.  Ans. 

2.  A  vessel  3.5  feet  each  side  within,  and  5  feet  deep,  to  find 
the  content.  Ans.  3.5x3.5=12.25x5  =  61.25  content. 

3.  A  piece  of  timber  is  1  foot  6  inches  broad,  9  inches  thick 
9  feet  6  inches  long ;   required  the  content. 

Ans.  1.5 X. 75=1. 125x9.5  =  10.6875  content. 

4.  What  is  the  solidity  of  a  cylinder  whose  height  is  121 
and  diameter  45.2!  Ans.  45.22x  .7854  X  121  =  192442.6. 

5.  The  Winchester  bushel  is  a  hollow  cylinder  18-^  inches 
in  diameter,  and  8  inches  deep ;  what  is  its  capacity  ? 

Ans,  18.52X  .7853982=268.8025  X  8=:2150.43  cubic  inches 


248 


MENSURATION. 


6.  There  is  a  cistern  under  ground,  in  the  form  of  a  parallel 
ogram,  in  length  16  feet,  width  12  feet,  depth  9  feet ;  required 
the  number  of  hogsheads  it  will  contain.  Thus  :  16x12x9= 
1728  cubic  feet  of  space,  then  17282=2985984  cubical  inches, 
which  divide  by  231,  the  number  of  inches  in  a  gallons  12926 
gallons— 63=205  hogsheads,  11|J  gallons.  Ans. 

7.  How  many  hogsheads  will  a  circular  cistern  contain 
which  is  7  feet  in  diameter,  and  9  feet  in  depth  ? 

Ans,  41  hogsheads,  7  gallons,  3  quarts,  1  pint. 

When  the  breadth  and  thickness  of  a  piece  of  timber  are  given  in 
inches,  to  find  how  much  in  length  will  make  a  solid  foot. 

RULE.  r^ 

Divide  1728  by  the  product  of  the  breadth  and  depth,  aiid  the 
quotient  will  be  the  answer  or  length,  making  a  solid  foot. 

In  a  piece  of  timber  1 1  inches  broad  and  8  inches  thick,  how 
many  inches  in  length  will  make  a  solid  foot  1 

Ans.  11x8  =  88)1728(19.6. 

To  find  how  many  solid  feet  a  round  stick  of  timber  of  equal  thick" 
ness  will  contain,  when  hewn  square, 

RULE. 

Multiply  twice  the  square  of  its  radius,  in  inches  by  the 
length  in  feet,  then  divide  the  product  by  144,  and  the  quotient 
will  be  the  answer. 

1.  Admitting  the  radius  of  a  round  stick  of  timber  to  be  11 
inches,  and  its  length  20  feet,  how  many  solid  feet  will  it  con- 
tain when  hewn'square  ?  Thus:  llXllX2x20=:4840~144 
=  33.6  solid  feet  of  hewn  timber. 

To  find  how  many  feet  of  square-edged  boards  of  a  given  thick' 
ness,  can  tm  sawn  from  a  log  of  a  given  diameter. 


Find  the  solid  content  of  the  log  when  made  square  by  the 
last  rule  ;  then  say,  as  the  thickness  of  the  board,  including  the 
saw-calf,  is  to  the  solid  foot  ::  so  is  12  inches  to  the  number  of 
feet  of  boards. 

1.  How  many  feet  of  square-edged  boards  1.25  inches  thick, 
including  the  saw-calf,  can  be  sawn  from  a  log  20  feet  long  and 
24  inches  in  diameter?  Thus:  12x12x2=288x20  =  5760 
-^144=40  feet  solid  content;  then,  1.25  :  40  ::  12  :  384  fee^ 
of  boards,  Ans. 


MENSURATION.  349 

Miscellaneous  Matter  in  Mensuration. 

1.  Purchased  a  box  of  window-glass  containing  256  panes  of 
glass,  8  by  10  inches  ;  required  the  number  of  square  feet,  and 
the  cost  at  5c.  per  foot. 

^715.  8X10  =  80  in.  :  256x80=20480-^144  =  142.22  square 
feet,  D7.11.1. 

2.  Bought  a  box  of  window-glass  containing  320  square  feet 
of  glass  10X12  inches  ;  required  the  number  of  panes. 

Ans.  10x12  =  120  :  320  X  144=46080-f- 120  =  384  panes. 

3.  In  a  pile  of  wood  7.5  feet  long,  Q\  feet  high,  and  cut  4.75 
feet,  how  many  cords,  and  how  much  will  it  come  to  at  D5  per 
cord?  Ans.  7.5x6.2.5x4.75=22265625^128  =  1.73950 
cords  xD5=8.69.75. 

4\  Fjom  a  mahogany  plank  26  inches  broad,  a  yard  and  a 
half  is  to  be  sawn  off;  wh^  distance  from  the  end  must  the  line 
be  struck?  ^^:i  IJ^^-f^  iw.fir  -^^  -/^  ■■::  y'-V^/^n^.  6.23  feet. 

5.  A  joist  is  8^  inches  3eep,  3.5  broad;  what  will  be  the 
dimensions  of  a  scantling  just  as  large  again  as  the  joist,  that  is 
4j  inches  broad?  Ans.  12.52  inches  deep. 

6.  How  many  3-inch  cubes  can  be  cut  out  of  a  12-inch  cube  ? 

Ans.  64. 

7.  A  may-pole,  whose  top  was  broken  off  by  a  blast  of  wind, 
struck  the  ground  at  15  feet  distance  from  the  bottom  of  the 
pole  ;  what  was  the  height  of  the  whole  may -pole,  supposing 
the  length  of  the  broken  piece  to  be  39  feet  ?  Ans.  75  feet. 

8.  The  diameter  of  a  circle  is  25  rods  ;  required  the  length 
of  a  stone  wall  that  will  enclose  it.  Ans.  78.54  rods. 

9.  How  many  men  may  stand  on  one  acre  of  land,  allowing 
12  square  feet  to  each  man  ?  Ans.  3630. 

10.  How  many  solid  yards  of  earth  will  be  thrown  out  to 
make  a  cellar  48  feet  long, 27  wide,  and  6  deep?  Ans.  288. 

11.  There  is  an  island  50  miles  in  circumference,  and  three 
men  start  together  to  travel  the  same  way  about  it;  A.  goes  7 
miles  per  day,  B.  8,  and  C.  9  ;  when  will  they  come  together 
again  ?  Thus,  50x7+50x8-f  50"x9"-f-7-|-8  +  9  = 
50  days,  time  of  meeting,  and  A.  will  travel  350  miles,  B.  400, 
C.  450.  A71S. 

12.  A  general  disposing  his  army  into  a  square  battalion, 
found  he  had  231  men  remaining,  but  increasing  each  side  with 
one  soldier,  he  wanted  44  to  fill  the  square  ;  of  how  many  men 
did  his  army  consist?  Thus:  2314-44=275  and 
275-1-7-2  =  137,  then  137x137+231=19000.  Ans. 

Proof,  138x138=19044  :  ^19044=138. 


250  3IENSURATI0N. 

13.  Suppose  a  lighthouse  built  on  the  top  of  a  rock  ;  the  dis* 
tance  between  the  place  of  observation  and  that  part  of  the  rock 
level  with  the  eye  is  620  yards  ;  the  distance  from  the  top  of 
the  rock  to  the  place  of  observation  is  846  yards  ;  and  from  the 
top  of  the  lighthouse,  900  yards  ;  the  height  of  the  lighthouse 
is  required. 


Ans.  y900x900-620x 620-^846x846-620x620  = 
76.77  yards. 

14.  Suppose  one  of  those  meteors  called  fire-balls,  to  move 
parallel  to  the  earth's  surface,  and  50  miles  from  it,  at  the  rate 
of  20  miles  per  second :  in  what  time  would  it  move  round  the 
earth  ?  

Thus  :  the  earth's  diameter  is  7964  miles,  then  7964  +  50  x  3 
=r8064=the  diameter  of  the  circle  described  by  the  ball, 
then  8064x3.1416=25333.8624  miles  its  circumference,  and 
25333.8624  '    ''    '''     ''''     '""     """     '""" 

-^20=1266.69312sec'ds=21   6  41     35     13       55       12  ^n5. 

1 5.  The  mean  distances  of  the  planets  from  the  sun, in  English 
miles,are  as  follows:  Mercury  36686617.5,  Venus  68552135.83, 
Earth  94772980  (or  95000000),  Mars  144404783.33,  Jupiter 
492912533.33,  Saturn  903957657.5  ;  now,  as  a  cannon-ball  at 
its  first  discharge  flies  a  mile  in  8  seconds,  and  sound  1142  feet 
in  a  second,  in  what  time,  at  the  above  rate,  would  a  ball  pass 
from  the  earth  to  the  sun,  and  sound  move  from  the  sun  to 
Saturn  1 

Thus:  94772980 X8''  =  758183840=24  years,  15  days, 
6  hours,  27  minutes,  20  seconds,  for  the  passage  of  the  ball.  And 
903957657.5  X  5280=4772896431600  feet,  and  4772896431 600 
-f-1142  =  132  years,  192  days,  ^i^hours,  42  minutes,  21|ff 
seconds,  for  the  passing  of  sound  from  the  sun  to  Saturn.  Ans. 

16.  Light  passes  from  the  sun  to  the  earth  in  8.2  minutes; 
in  what  time  would  it  pass  from  the  sun  to  the  Georgium  Sidus 
(Herschel),  it  being  1803930416.66  English  miles? 

Ans.  As  94772980  :  8.2  ::  1803930416.66  :  2  h.  36  m.  4"  50'^'. 

17.  If  a  cubic  foot  of  iron  were  hammered  or  drawn  into  a 
square  bar,  an  inch  about,  that  is,  \  of  an  inch  square,  required 
its  length,  supposing  there  is  no  waste  of  metal. 

Thus,  12  X  12  X  r2-f-.25  X  .25  X  4  =  6912  inches  =  576  ft.  Ans. 

18.  How  many  square  feet  of  boards  in  a  load  consisting  of 
150,  12^  fe^tln  length,  and  11  inches  in  width? 

Thus:  12.5x11  =  137-^12  =  11.46  feet  in  one  board;  11.46 
X  150=1719  square  feet  (nearly),  Ans. 


MBNSURATION'.  25a 

19.  The  solid  content  of  a  square  stone  is  found  to  be  136^ 
feet,  its  length  is  9^  feet ;  what  is  the  area  of  one  end  ?  and  if 
the  breadth  be  3  feet  11  inches,  what  is  the  depth? 

^,  136.5X1728  on«on^or   •  a    2069.0526 

Thus: ==  area  2069.0526  m.    and 

9.5x12  47 

=  44.022  inches.  Ans. 

20.  How  many  perches  of  stone  are  there  in  a  wall  48.25 
feet  in  length,  1.5  in  width,  and  12  feet  in  height? 

Ans.  35.09+  perches, 

21.  Required  the  quantity  of  wood  in  a  tier  64.5  feet  in 
length,  4  feet  in  width,  and  4  feet  in  height. 

Ans.  8.0625  =  83^1^  cords. 

22.  In  a  pile  of  stone  25  feet  in  length,  12j  feet  in  width, 
and  5i  feet  in  height ;  required  the  cubical  feet,  the  number  of 
perches,  and  the  value  at  75c.  per  perch  ? 

Ans.  1718.75  cubical  feet=69.444  perches,  value  D52.08.3-f- 

23.  How  much  wood  in  a  pile  14^  feet  in  length,  6 J  feet 
high,  and  3^  feet  in  width  ;  and  value  at  D4i  per  cord  ? 

Ans.  2.47802  cords  ;  Dll.15.1090  value. 

24.  What  is  the  breadth  of  a  piece  of  cloth  which  is  36 
yards  long,  and  which  contains  63  square  yards  ? 

Ans.  63^36  =  If  yards. 

25.  How  many  yards  of  carpeting  1^  yards  wide  will  cover 
a  iloor  30  feet  long  and  22-^  feet  broad  ? 

Thus:  30x22.5=1:675  ft.-r9  =  75  yds. -f-ll=60  yds.  Ans. 

26.  What  is  the  side  of  a  square  piece  of  land  containing  289 
square  rods  ?  ^^/^     ^j>.  i  i 

27.  If  the  diameter  of  the  earth  be  7930  miles,  what  is  the 
circumference?  .  ^71^.24912.8. 

28.  How  many  miles  does  the  earth  move  in  revolving  round 
the  sun,  supposing  the  orbit  to  be  a  circle  whose  diameter  is 
190y{nillions  of  miles  ?  Ans.  59690S«^ 

29.  What  is  the  circumference  of  a  circle  whose  diamefeVis 
769843  rods?  /  f  or)  Ij  9>^//^/h>l  V)^ 

30.  If  the  circumference  of  the  sun  be  2800000  miles,  what    j^ 
is  the  diameter?  Ans.  891267.^^* 

31.  What  is  the  diameter  of  a  tree  which  is  5^-  feet  round  ?  f^'^^ 

32.  If  the  circumference  of  the  moon  be  6850  miles,  what  is 
her  diameter?  ^w^.  2180  miles. 

33.  If  the  whole  extent  of  the  orbit  of  Saturn  be  5650  mill- 
ons  of  miles,  how  far  is  he  from  the  sun  ?di?,jy  /  7  9  ^,^^1^4 

134.  What  is  the  area  of  a  circle  whose  diameter  is  623  feet : 
Ans.  304836  square  feet 
I 


252  MENSURATION. 

To  measure  stone  in  a  well. 

In  taking  the  diameter  of  the  wall,  measure  from  centre  to 
centre,  or  to  the  clear;  add  the  thickness  of  the  wall;  and 
when  the  diameter  is  taken,  or  obtained,  find  the  circumference, 
which  is  the  length  of  the  wall. 


As  7  :  is  to  22  ::  so  is  the  diameter  (taken)  :  to  the  circumfer 
ence  required ;  having  found  the  length  of  the  wall,  and  meas- 
ured its  thickness  and  depth,  it  is  then  prepared  for  calculation 
as  explained  above. 

35.  A  well  walled  with  stone  1  foot  2  inches  thick,  its  diam- 
eter in  the  clear  2  feet  4  inches,  and  depth  50  feet ;  required 
the  length  of  the  wall,  and  the  quantity  of  stone. 

As  7  :  22  ::  3.5  :  11  feet  circumference,  or  length;  then  1  ft. 
2  inches  =  l. 1666  +  6  the  thickness,  11  length. 
50  depth.  1.1666  +  6 

11 


12.8332 
50 


24.75)641.66(25.925+ 


Is  what  the  well  measures  in  the  clear  (inside  the  walls) ;  add 
the  thickness  of  the  wall,  and  you  have  the  diameter  of  the  wall 
or  circle.  Ans.  25.925+  perched. 

36.  Required  the  number  of  perches  of  stone  in  a  well,  whose 
diameter  is  4  feet  9  inches  in  the  clear,  the  depth  of  the  well  20 
feet,  and  the  thickness  of  the  wall  15  inches. 

Ans.  19.047  perches,  or  19+  perches. 

37.  How  many  acres  are  there  in  a  circular  island  whose 
diameter  is  124  rods  ?  Ans.  75  acres,  76  rods. 

38.  If  the  diameter  of  a  circle  be  113  and  the  circumference 
355,  what  is  the  area  ?  Ans.  10029. 

39.  What  is  the  diameter  of  a  circle  whose  area  is  380.1338 
feet  ? 

Thus:  380.1336-^.7854=484  and  ^484=22.  Ans, 

40.  What  is  the  area  of  a  square  inscribed  in  a  circle  whosQ 
area  is  159?  Thus  :  .7854  :  \  ::  159  :  101,22,    Am 


MENSURATION.  253 

41.  What  is  the  side  of  a  square  whose  area  is  equal  to  that 
of  a  circle  452  feet  in  diameter  1 

Ans.  V  (452)?'x. 7854 =400.574 

42.  What  is  the  diameter  of  a  circle  which  is  equal  to  a  square 
whose  side  is  36  feet  ?  Ans.  ^/(36)2 ^0.7854  =  40.6217. 

43.  If  the  height  of  a  square  prism  be  2^  feet,  and  each  side 
of  the  base  lO^,  what  is  the  solidity? 

Ans.  the  area  of  the  base  lOi  x  10^=  106 J  square  feet ;  and 
the  solid  content  =106jx2i=:240|  cubic  feet. 

44.  How  many  gallons  ale  are  there  in  a  cistern,  which  is  11 
feet,  9  inches  deep,  and  whose  base  is  4  feet,  2  inches  square  ? 

The  cistern  contains  352500  cubic  inches-f-282  =  1250  gal- 
lons. Ans. 

45.  Suppose  a  cellar  dug  16  feet  in  length,  16  in  width,  and 
6  feet  in  depth  ;  required  the  quantity  of  stone  to  enclose  a  wall 
1.5  feet  thick,  and  6  feet  in  height,  in  side  of  said  cellar. 

Ans.  21i,  perches.  (24f  ft.  =  l  perch.) 

46.  A  room  is  11  feet  high,  21  feet  long,  and  16  feet  wide ; 
how  many  cubic  feet  of  space  in  it  ?  Ans.  3696  cubic  feet. 

47.  How  many  square  feet  in  a  stack  of  15  boards,  12  feet, 
8  inches  in  length,  and  13  inches  wide  ? 

Ans.  205^=205  ft.  10  in. 

48.  Suppose  a  ship  sails  from  latitude  43°,  north,  between 
north  and  east,  till  her  departure  from  the  meridian  be  45  leagues, 
and  the  sum  of  the  distance  and  difference  of  latitude  be  135 
leagues  ;  I  demand  her  distance  sailed  ?  and  latitude  come  to  ? 

Ans.     135 X 135-45 x45^gQ    ^  ^^^    60x3  =  180 

135X2 
miles  =  3  degrees,  the  difference  of  latitude  135  —  60=75 
leagues,  the  distance.  Now,  the  vessel  is  sailing  from  the  equa- 
tor, and  consequently  the  latitude  is  increasing ;  therefore,  to  the 
latitude  sailed  from  43°  add  the  difference  of  latitude  3°,  and 
the  sum  is  the  latitude  come  to,  that  is,  46  degrees. 

49.  Four  men,  E,  F,  G,  and  H,  purchased  a  grindstone  of  60 
inches  in  diameter ;  how  much  of  its  diameter  must  each  grind 
off,  to  have  an  equal  share  of  the  stone,  if  one  first  grind  his 
share,  and  then  another,  till  the  stone  is  ground  away,  making 

no  allowance  for  the  eye.  ^^  ^:  ^2^:6^'=>^/V(^.Of^Z,^^ai'^'^  >Vi^/ 

ak  JqucL>^  M- S./9^^^9      RULE.^ ^'^      ^  ' mi 

Divide  the  square  of  the  diameter  by  the  number  of  men , 
subtract  the  quotient  from  the  square,  and  extract  the  square 
root  of  the  remainder,  which  is  the  length  of  the  diameter  after 


254 


MENSURATION. 


the  first  man  has  ground  his  share  ;  this  work  being  repeated  by 
subtracting  the  same  quotient  from  the  remainder,  for  every  man 
to  the  last ;  extract  the  square  root  of  the  remainder,  and  sub- 
tract those  roots  from  the  diameter,  one  after  another,  and  the 
several  remainders  will  be  the  answers. 


Thus  : 


60  in.  diam, 
60  X 


Again :  60  in.  diam.  of  stone. 
—■51.9615 


4)3600 


Quot.  900 


from  3600 
take    900 


2/2700  >  51.9615- 
jL  900  (  —    60 


8.0385= 1st,  or  A.'s  share. 


51.9615 
—42.4264 


9.5351z=2d,  or  B.'s  share. 


42.4264 
-30.0000 


2/1800  >  =42. 
JL  900  J— 51. 


J  900  >  =30.1 
^  S  —42. 


4264—  12.4264  =  3d,  C.'s  share. 

9615  and  the  remaining 

30  in.  is  D.'s  share. 

=  30.000—  for  8.0385+9.5351  +  12.4264 

4264  +30. =60  in.  the  dia.of  the  st 


Or,  geoniettically,  thus  :  on  the  radius  A  B  describe  a  semi-  1 
circle;  al^o  divide' A  B  into  as  many  equal  parts  as  there  are 
shares,  and  draw  perpendiculars  from  the  points  of  division  to 
the  semicircle  on  A  B  ;  then  with  the  centre  B,  and  radii  B  D^ 
B  C,  &c.,  describe  circles,  and  the  question  is  solved. 


MENSURATION. 


255 


50.  If  a  load  of  wood  is  8  feet  long  and  3  feet  wide,  how 
high  must  it  be  to  contain  1  cord  ?  Ans.  51  ft.  =  5  ft.  4  in. 

51.  Required  the  quantity  of  stone  in  a  pile  16.5  feet  in 
length,  4.5^  feet  in  width,  and  4  feet  in  height,  computed  at  16.5 
cubic  feet  per  perch.  Ans.  18  perches. 

52.  If  the  floor  of  a  square  room  contain  36  square  yards, 
how  many  feet  does  it  measure  on  each  side  ?         Ans.  18  feet. 

53.  What  is  the  difference  between  100  square  feet  and  100 
feet  square  1  Ans.  9900. 

54.  Suppose  a  ladder  60  feet  long  so  planted  as  to  reach  a 
window  37  feet  from  the  ground,  on  one  side  of  the  street,  and 
without  moving  it  at  the  foot,  will  reach  a  window  23  feet  high, 
on  the  other  side  ;  required  the  breadth  of  the  street  1 

Ans.  102.64. 


55.  The  Bunker  Hill  monument 
in  Charlestown,  near  Boston,  is 
said  to  be  225  feet  high ;  the  as- 
cent is  gained  by  winding  stairs  on 
the  inside,  and  also  by  steam-pow- 
er, acting  on  machinery  for  that 
purpose  ;  suppose  a  person  stand- 
ing 121  feet  from  the  base,  and 
the  ground  level,  what  length  of 
cord  would  be  required  to  reach 
from  the  ground  to  the  top  of  the 
monument,  allowing  the  side,  or 
wall  to  be  perpendicular  (which, 
however,  is  not  the  case). 

Thus:  225x225  =  50625;  12X2 
=  14641  +  50625  =  65266;  then 
y/65266 =255.47  feet  length  of 
cord.  Ans. 


56.  Required  the  length  of  a  rope  that  will  be  sufficient  for  a 
horse  to  graze  just  2  (circular)  acres  ? 

Thus  :  2  A.  (y)  =  9680  yards  =  98.3869  + root,  then  98.3869 
X1.12837  =  lli:il6826353  yards,  whole  diameter  which-~2 
=  55.55  +  yards.  Ans.  '  % 

57.  What  is  the  side  of  a  square  piece  of  land  containing 
124J  acres  ?  Ans.  141  rods 


256  MENSURATION. 

58.  The  height  of  a  tree  growing  in  the  centre  of  a  circula? 
island  44  feet  in  diameter,  is  75  feet,  and  a  line  stretched  from 
the  top  of  it  over  the  hither  edge  of  the  water,  is  256  feet ;  what 
is  the  breadth  of  the  stream,  provided  the  land  on  each  side  of 
the  water  be  level  ? 

Thus:  256x256  =  65536;  and  75 X 75=5625  and ^59911 
=244.76  +  and  244.76 -f^=  (22)  =::222.76  feet.  Ans. 

59.  What  is  the  side  of  a  square  field  which  contains  58f 
acres  ?  ^^ns     q(.(^i- 

60.  What  i«  the  width  of  a  piece  of  land  which  is  280  rods 
long,  and  which  contains  77  acres  ? 

Ans.  77 X  160-12320—280=44  rods. 

61.  The  Winchester  bushel  is  a  hollow  cylinder,  18^  inches 
in  diameter,  and  8  inches  deep  ;  what  is  its  capacity  ? 

Ans.  2150.42. 

62.  What  is  the  solidity  of  a  cylinder,  whose  height  is  424, 
and  circumference  213?  Ans.  1530791-}- 

63.  What  is  the  convex  surface  of  a  right  cylinder,  which  is 
42  feet  long,  15  inches  in  diameter  ?  . 

Ans.  42x1.25x3.14169  =  164.933  square  feet. 

64.  What  is  the  solidity  of  a  cylinder,  whose  height  is  121, 
and  diameter  45.2?  A71S.  45.22  x  .7854  x  121  =  194156.6. 

65.  What  is  the  square  root  of  37|f  ? 

Ans.  37ff  x49-|-36=:if|9y=6f 

66.  In  a  circular  cistern,  whose  diameter  at  bottom  is  9  feet, 
and  top  7  feet,  depth  12  feet,  required  the  number  of  hogsheads 
it  will  contain.  Thus:  8x8=64  square,  mean  diameter, 
X. 7854=50.2656  area  base,  or  mean  diameter,  X  12,  height  = 
603.1872,  content  in  feet,  x  1728  =  1042307.4816  cubic  inches, 
-^231=4512iff  gallons-^63=71  hogsheads,  39  gallons,  2-f- 
quarts.  Ans. 

2d.  As7  :  8  ::  22  :  25.1428-4-2  =  12.5714x4,  half  diameter, 
=  50.2856x12,  length,  =603.4272X1728  =  1042722.2016-^ 
231=4513.9489  gallons,  -^63=71  hhds.,  40  galls. ^-   Ans. 

67.  Boston  is  6**  40'  east  longitude,  from  the  city  of  Wash- 
ington ;  when  it  is  6  o'clock,  P.  M.,  at  Washington,  what  is  the 
hour  at  Boston?  Thus:  6x4=24'  :  40x4=160''=2' 40'^ 
-h24'=26^  40'''  past  6.  Ans. 

68.  How  many  planks  15  feet  long,  15  inches  wide,  will  floor 
n  barn  which  is  60^  feet  long,  and  33^  wide  ?         Ans.  108/^. 

69.  If  we  are  95  millions  of  miles  from  the  sun,  and  if  the 
earth  revolves  round  it  in  365J  days  ;  how  far  are  we  carried 
in  24  hours  ?  Ans.  1  million  634  thousand  miles. 


^  MENSURATION.  257 

V  70.  If  the  diameter  of  Saturn's  larger  ring  be  205000  and 
^i90000  miles,  how  many  square  miles  on  one  side  of  the  ring? 
^    Thus  :  395000  x  15000  x  .7854=4653495000  miles.  Ans. 

*'  71.  What  is  the  expense  of  paving  a  street  20  rods  long  and 
^2  rods  wide,  at  5  cents  for  a  square  foot?  Ans.  D544|-. 

>$"  72.  How  many  hills  of  corn  may  be  planted  on  an  acre,  3  ft. 
'^distance?  Thus:   1  acre  ==43560  square  feet^ (3x3 =9) 

^=4840  hills,  Ans. 

s^  73.  How  many  trees  may  be  planted  on  10  acres,  at  6  feet 
^distance?  Ans.    12100. 

^  74.  How  many  rails  will  it  require  to  enclose  a  field  of  16 
"Nacres?  (length  of  rails  13  feet,  and  6  rails  high.)  v  ^  ; 

^     75.  What  is  the  side  of  a  square  whose  area  is  equal  to  that 

>  of  a  circle  452  feet  in  diameter  ?  ^^  ^  ^  '^  i  I 

;;;  Ans.  y^ (452)2 X. 7854=400.574. 

^  76.  What  is  the  diameter  of  a  circle  which  is  equal  to  a 
vj square  whose  side  is  36  feet?  Ans,  ^(36)2-^0.7854  =  40.6217. 
':^  77.  If  a  carriage-wheel  4  feet  in  diameter  revolve  300  times 
^  in  going  round  a  circular  green,  what  is  the  area  of  the  green  ? 
":«  Ans.  4154  square  rods,  or  25  acres,  3  qrs.,  34  rods. 

V  78.  What  is  the  solidity  of  a  wall  which  is  22  feet  long,  12 
>?  feet  high,  and  2  feet  6  inches  thick  ?  Ans.  660  cubic  feet. 
^  79.  If  the  height  of  a  square  prism  be  1\  feet,  and  each  side 
o  of  the  base  10^  feet,  what  is  the  solidity? 

o  Ans.  106|-  square  feet,  240^  cubic  feet. 

^      80.  How  many  ale  gallons  are  there  in  a  cistern  which  is  11 
feet  9  inches  deep,  and  whose  base  is  4  feet  2  inches  square  ? 
Ans.  (-^282)  352500  cubic  inches  =  1250  gallons. 

81.  The  highest  point  of  the  Andes  is  about  4  miles  above 
the  level  of  the  ocean.  If  a  straight  line  from  this  touch  the 
surface  of  the  water  at  the  distance  of  178J[  miles,  what  is  the 
diameter  of  the  earth  ? 

Thus:   178.252^4m.=7939.2656  miles,  Ans, 

82.  If  the  diameter  of  the  earth  be  7939.2656  miles,  and 
Mount  Etna  2  miles  high,  how  far  can  its  summit  be  seen  at 
sea?  Thus:  7939.2656x2  miles+22=  15882  ; 
then  y  15882.5312  =  126-1-  miles.  Ans. 

83.  How  much  water  can  be  put  into  a  cubical  vessel  3  feet 
deep,  which  has  been  previously  filled  with  cannon-balls  of  the 
same  size,  2,  4,  6,  or  9  inches  in  diameter,  regularly  arranged 

in  tiers,  one  directly  above  another?       Ans.  96^  wine  gallons .^^/c^^U 

84.  How  many  such  globes  as  the  earth  are  equal  in  bulk  to 
the  sun,  if  the  former  is  7930  miles  in  diameter,  and  the  latter     . 
890000?  Ans.  1413678,^/*^^ 

22* 


258  PROMISCUOUS  QUESTIONS. 


What  is  the  rule  to  measure  wood  ?  What  is  a  square  ? 
How  will  you  find  the  area  of  a  square  having  equal  sides  ? 
What  is  a  rectangle  ?  How  will  you  find  the  area  of  a  rectan- 
gle ?  What  is  a  triangle  ?  How  will  you  find  the  area  of  a 
triangle  ?  What  is  a  circle  ?  What  is  the  diameter  of  a  circle  ?' 
W^hen  the  diameter  is  given,  how  will  you  find  the  circumfer- 
ence ?  When  the  circumference  is  given,  how  will  you  find 
the  diameter  ?  How  can  you  find  the  area  of  a  circle  ?  If  the 
area  is  given,  how  will  you  find  the  diameter  ?  how  the  circum- 
ference ?  What  is  a  sphere?  How  will  you  find  the  surface, 
of  a  sphere  ?  How  will  you  find  the  solidity  of  a  sphere  ?' 
What  is  a  prism  ?  How  will  you  find  the  solidity  of  a  prism  ? 
What  is  a  cylinder  1  How  will  you  find  the  convex  surface  of 
a  cylinder?  How  will  you  find  the  solidity  of  a  cyhnder? 
What  is  a  cone  ?  How  will  you  find  the  solidity  of  a  cone  ? 
What  is  a  pyramid  ?  How  will  you  find  the  solidity  of  a 
pyramid  ?  What  is  a  parallelopipedon  ?  How  will  you  find 
the  solidity  of  a  parallelopipedon  ?  What  are  the  rules  for 
measuring  timber,  &c.  ? 


PROMISCUOUS   QUESTIONS. 

1.  What  is  the  interest  of  D650.75  for  150  days  at  7  per 
cent.?  Ans.  Dl8.72.r 

2.  What  is  the  interest  of  D440  for  30  days  at  6  per  cent.  ?      j 

Ans.  D2.17.^ 

3.  Required  the  interest  of  D212  for  60  days  at  5  per  cent. 

Ans.  m. 74,2.^ 

4.  A.  sells  goods  on  commission  at  2.5  per  cent. ;  how  much 
will  he  receive  for  selling  D1800  worth?  Ans.  D4.50. 

5.  How  much  must  a  broker  receive  for  discounting  D2000 
at  3.5  per  cent.  ?  Ans.  D70. 

6.  What  is  the  commission  on  D2176.50  at  2.5  per  cent.  ? 

^7^5.  D54.41.2. 

7.  What  is  the  premium  of  insuring  D1650  at  15.5  per  ct.  ? 

Ans.  D255.75. 

8.  How  many  bushels   of  wheat   can   be   purchased   with 
D81.76  at  D1.12  per  bushel  ?  Ans.  73. 


PROMISCUOUS    QUESTIONS.  269 

9.  How  much  will  27c wt.  of  iron  cost  at  D4.56  per  cwt.  ? 

Ans.  D123.12. 

10.  Purchased  a  quantity  of  goods  for  D250,  and  3  months 
after  sold  them  for  D275 ;  how  much  per  cent,  per  annum  was 
gained  ?  Ans.  40  per  cent. 

11.  In  what  time  will  a  sum  double  itself  at  6  per  cent,  sim- 
ple interest?  .  Ans.  16  years,  8  months. 

12.  Required  the  amount  of  the  following  bill,  namely:  150 
bushels  of  rye  at  Dl  ;  200  bushels  of  wheat  at  Dl.60  per  bush- 
el ;  50  bushels  of  com  at  80c.  per  bushel ;  20  bushels  of  oats 
at  60c.  per  bushel ;  40  bushels  of  salt  at  40c.  per  bushel ;  10 
bushels  of  potatoes  at  20c.  per  busliel ;  3.5  bushels  of  flax-seed 
at  Dl.25  per  bushel ;  5  bushels  of  cloverseed  at  D5  per  bushel ; 
15  bushels  of  barley  at  Dl.25  per  bushel ;  4  bushels  of  turnips 
at  30c.  per  bushel ;  8  bushels  of  onions  at  25c.  per  bushel ;  2 
bushels  of  beans  at  Dl.25  per  bushel ;  9  bushels  of  apples  at 
75c.  per  bushel.  Ans.  D600.57.5. 

13.  What  is  the  amount  of  the  following  bill,  namely :  4  parrs 
of  boots  at  D5  per  pair ;  6  pairs  of  small  shoes  at  75c.  per  pair ; 
7  yards  of  muslin  at  20c.  per  yard ;  150  lbs.  of  flax  at  20c.  per 
pound  ;  4c wt.  of  flour  at  D3  per  cwt, ;  50  bushels  of  malt  at  Dl 
per  bushel ;  70  bushels  of  apples  at  20c.  per  bushel ;  50  lbs.  of 
sugar  at  12.5c.  per  pound ;  208  bushels  of  potatoes  at  30c.  per 
bushel ;  171  bushels  of  corn  at  Dl  per  bushel ;  43  lbs.  of  beef 
at  7c.  per  pound ;  324  yards  of  linen  at  50c.  per  yard  ? 

Ans.  D536.56. 

14.  What  is  the  weight  and  value  of  13  bags  of  wheat,  which 
weigh  as  follows,  at  D1.35  per  bushel,  at  601bs.  per  bushel? 

No.  1.  leOlbs.  ;  No.  2,  165lbs. ;  No.  3,  172lbs.  ;  No.  4,  173 

lbs.  ;    No.  5,  ISOlbs.  ;    No.  6,  1691bs. ;  No.  7,  195lbs.  ;  No.  8, 

185lbs. ;  No.  9, 184lbs  . ;  No.  10, 1631bs. ;  No.  11,  1701bs. ;  No. 

No.  12, 1901bs. ;  No.  13,  1671bs.  (the  weight  of  the  sacks  261bs.) 

fi;-  g^^<l'}X  ^^)^^h^'\^Q'^^  ^"^'  22731b.,  value  D50.55.7. 

15.  How  much  will  trie  following  bill  amount  to,  namely: 
120  bushels  of  apples  at  20c.  per  bushel ;  110  bushels  of  rye  at 
Dl  per  bushel ;  50  bushels  of  salt  at  40c.  per  bushel ;  70 
bushels  of  sea-coal  at  30c.  per  bushel ;  30  bushels  of  malt  at 
50c.  per  bushel ;  40  bushels  of  peas  at  30c.  per  bushel ;  30 
bushels  of  beans  at  40c.  per  bushel ;  80  bushels  of  oats  at  70c. 
per  bushel;  90  bushels  of  wheat  at  Dl.60  per  bushel;  100 
bushels  of  corn  at  70c.  per  bushel ;  80  bushels  of  potatoes  at 
80c.  per  bushel ;  and  40  bushels  of  turnips  at  20c.  per  bushel  ? 

Ans,  D516.00. 


260  PROMISCUOUS  QUESTIONS. 

16.  What  is  the  value  of  1403  lbs.  of  wheat  at  Dl.25  pel 
bushel  of  60  lbs.  ?  Ans.  D29.22.9. 

17.  If  the  earth  be  360  degrees  in  circumference,  and  each 
degree  69.5  miles,  what  time  would  it  require  for  a  person  to 
travel  round  the  globe  at  the  rate  of  20  miles  per  day,  and  th© 
year  to  consist  of  365.25  days?  Ans.  3  years,  155.25 days 

18.  A  line  35  yards  long  will  exactly  reach  from  the  top  of  a 
fort  standing  on  the  brink  of  a  river,  to  the  opposite  bank,  known 
to  be  27  yards  broad ;  required  the  height  of  the  wall. 

Ans.  22  yards,  9.^  inches. ^^ 

19.  How  mTk;ii  will  27cwt.  of  sugar  come  to  at  9c.  per  lb.  ? 

Ans.  D272.16. 

20.  How  mucn  will  281  yards  of  tape  cost  at  9m.  per  yard? 

Ans.  D2.52.9. 

21.  What  is  the  cost  of  29.5  yards  of  flannel  at  75c.  per  yd.  ? 

Ans.  D22.12.5. 

22.  How  much  will  371  yds.  of  broadcloth  come  to  at  D5.79 
per  yard?  ^n^.  D2148.09.0. 

23.  If  a  man  receive  yearly  D949,  how  much  is  it  daily? 

Ans.  D2.60. 

24.  How  much  will  32625  feet  of  boards  come  to  at  D8.25 
per  M.?  Ans  1)269.15.6+ 

25.  At  D5.50  per  M.  what  will  21186  feet  of  boards  come 
to?  Ans.  1)116.52.3. 

26.  When  boards  are  sold  at  D18  per  M.  how  much  is  it  per 
foot?  Ans.  Ic.  8m. 

27.  The  sale  of  certain  goods  amount  to  D1875  40c. ;  what 
sum  is  to  be  received  for  them,  allowing  2.5  per  cent,  for  com- 
mission, and  .25  for  prompt  payment  of  the  net  proceeds  ? 

Ans.  D1823.82.65. 

28.  If  4.5  yards  of  broadcloth  cost  D25,  what  will  17.25 
yards  cost?  Ans.  D95.83.3 

29.  If  30  yards  of  cloth  cost  DUO,  how  much  is  it  per  yd.  ? 

AnsfD3.66.6,   " 

30.  A  farmer  purchased  goods  at  a  store  to  the  amount  of 
D180  ;  he  is  to  pay  one  half  in  wheat  at  Dl.25  per  bushel,  and 
the  other  half  in  rye  at  87.5cts.  per  bushel ;  required  the  quan- 
tity of  each  kind. 

Ans.  wheat  72  bushels,  rye  102  bush.,  3  pecks,  3+  quarts. 

31.  When  wheat  is  selling  at  Dl.60  per  bushel,  what  is  the 
price  of  1  quart  ?  Ans.  5c. 

32.  Paid  D140  for  2000  lbs.  of  cheese;  how  much  did  it 
cost  per  lb.  ?  and  if  I  sell  it  for  2c.  per  lb.  more  than  the  cost, 
what  is  my  profit?  Ans.  cost  7c.  per  lb. ;  and  profit  D40. 


PROMISCUOUS  QUESTIONS.  261 

33.  If  I  give  D9.50  for  2  lb.  of  indigo,  how  much  do  I  pay 
per  ounce,  and  how  much  do  I  gain  or  lose  by  selling  it  at  25c, 
per  ounce,  avoirdupois  ? 

Ans.  paid  29.6c.,  and  lost  Dl.50  on  the  whole. 

34.  If  I  lost  Dl.50  on  the  above  indigo,  at  what  price  per 
oz.  should  I  have  sold  it  to  gain  Dl.50,  and  at  what  rate  per 
cent,  profit  would  it  be  1 

Thus  :  D9. 50+3.00,  gain  on  2  lbs.  =  Dl2.50-i-32oz.=2  lb. 
=  39+  should  have  sold  for  ;  then  9.50  :  1.50  ::  100  :  15.7+ 
gain  per  cent.  Ans. 

35.  How  much  is  the  difference  between  the  interest  and 
discount  of  D500  for  1.5  years,  at  6  per  cent.  ? 

Ans.  D3.71.54 

36.  What  is  the  interest  of  D480,  for  60  days,  at  5  per  cent.? 

Ans.  D3.94.5+ 

37.  A.  loaned  B.  DlOO,  at  5  per  cent.,  compound  interest, 
and  to  C,  DlOO  at  10  per  cent.,  simple  interest;  payment  is  to 
be  made  when  the  interest  of  the  two  sums  shall  be  equal ;  re- 
quired the  interest  and  time. 

(By  approximation.)  Ans.  interest  D265.70  on  each  princi- 
pal ;  time,  26  years,  208  days. 

38.  What  is  the  square  root  of  368863  ? 

Ans,  607.34092  + 

39.  What  is  the  cube  root  of  .467764060?  c^mi'^^C  -f- 

40.  If  4.44  bushels  of  wheat  cost  D7.50,  what  will  17.22 
bushels  cost  ?  Ans.  D29.08.7. 

41.  If  7.5  pounds  of  coffee  cost  65c.,  what  will  1 12  lbs.  cost  ? 

Ans.  D9.70.6. 

42.  What  is  the  cost  of  4  hogsheads  of  wine,  at  9c.  per  pint! 

^n^.  D181.44. 

43.  If  a  man  travel  96  miles  in  3  days,  how  many  could  he 
travel  in  3  weeks  ?  Ans.  672. 

44.  If  675  yards  cost  Dl2,  8d.  2c.,  5m.,  how  many  yards 
may  be  had  for  38m.  ?  Ans.  2  yards. 

45.  If  a  yard  of  riband  cost  4.5c.,  how  much  will  it  require 
to  purchase  345  yards  ?  Ans.  D15.52.5. 

46.  If  19  yards  of  sheeting  cost  D25,  7d.  5c.,  what  will 
435.5  cost  ?  Ans.  D590,  2d.  Ic.  7m. 

47.  What  is  the  value  of  .3375  of  an  acre  ? 

Ans,  1  R.  14  poles. 

48.  What  is  the  value  of  .3  of  a  year  ? 

Ans.  109  days,  13  hours,  48  minutes. 

49.  What  is  the  value  of  .6875  of  a  yard  ? 

Ans.  2  qrs.  3  nail* 


263  PROMISCUOUS  QUESTIONS, 

50.  What  is  the  value  of  .76  of  a  league  ? 

Ans,  2  m.  2  fur.  9  po.  3  yd.  3  inches. 

51.  A  person  purchased  10  cords  of  wood  ;  on  measuring  it, 
he  found  it  3  inches  too  short,  which  should  have  been  4  feet ; 
now  much  did  he  lose  in  10  cords  ? 

Length,  4  ft.  =  48  inches^3  =  16  or  J^  per  cord  too  short; 
128-^-16  =  8  ft.  in  1  cord;   10x8  =  80  feet.  Ans. 

Or,  96x48x3  =  13824-^1728  =  8x10  =  80  feet.  Ans. 

52.  Required  the  quantity  of  wood  in  a  parcel  28  feet  long, 
4  feet  high,  and  cut  3  feet  ? 

^71^.  28x5x3-f-128  =  3  cords,  1  qr.  4  feet. 

53.  If  5  men  can  reap  52.2  acres  in  6  days,  how  many  men 
will  it  require  to  reap  417.6  acres  in  12  days  ?      Ans.  20  men. 

54.  If  a  ceiling  be  60  feet,  3  inches,  by  25  feet,  6  inches,  re- 
quired the  number  of  square  yards. 

Ans.  60.25x25.54-9  =  170  yards,  6  feet. 

55.  Add  I  of  a  yard  to  f  of  an  inch. 

A?is.  Ill  yards,  or  14j\  inch. 

56.  Add  J  of  a  week,  ^  of  a  day,  and  -^  of  an  hour,  together. 

Ans.  2  days,  14^  hours. 

57.  Add  1|-  miles,  -^q  furlongs,  and  30  rods,  together. 

Ans.  1  mile,  3  furlongs,  18  poles. 

58.  Add  |-  of  a  cwt.,  ^^  of  a  lb.,  13  oz.,  and  ^  of  a  cwt.,  6 
lb.  together?  Ans.  1  cwt.  1  qr.  27  lb.  13  oz. 

59.  How  much  is  the  difference  between  |  of  a  rod  and  f  of 
an  inch?  Ans.  10  feet,  11^  inches. 

60.  How  much  is  the  difference  between  ^^  of  a  hogshead, 
and  j^g  of  a  quart  ?  Ans.  16  galls.  2  qts.  1  pt.  3/^  gills. 

61.  From  f  oz.  take  |-  pwt.  Ans.  ih  pwt.  3  grs 

62.  Required  the  product  of  6,  X  by  f  of  5.  Ans.  20. 

63.  Required  the  product  of  3f  X  4l|  1  Ans,  li^^. 

64.  What  will  3|  boxes  of  raisins  cost,  at  D2^  per  box  1 

Ans.  D8y7^. 

65.  If  I  of  a  yard  of  cloth  cost  D3,  what  is  the  price  pev 
yard  ?  Ans.  DSf 

66.  If  4|  gallons  of  molasses  cost  D2f ,  how  much  is  it  pev 
quart  ?  -4n^.  15ii^  cents. 

67.  If  1|  hogshead  of  wine  cost  D250J,  how  much  is  the 
wine  per  quart  ?  Ans.  85.3c.  per  qt. 

68.  What  number  is  that,  to  which  if  you  add  -^  of  f  of  itself, 
the  sum  will  be  39  ?  Ans.  30. 

69.  What  number  is  that,  which  being  multiplied  by  |,  the 
product  will  be  3^  ?  Ans,  3||^ 


PROMISCUOUS    QUESTIONS.  263 

Black  Board. 

1.  Reduce  3658  inches  to  yards. 

2.  Reduce  6490  grains  to  pounds. 

3.  Reduce  32044  scruples  to  pounds. 

4.  Reduce  108910592  drams  to  tons. 

5.  Reduce  59  m.  7  fur.  38  poles,  to  poles. 

6.  In  3278  nails,  how  many  yards  ? 

7.  In  19  A.  2  R.  37  po.  how  many  square  poles  ? 

8.  In  175  square  chains,  how  many  square  rods  ? 

9.  In  55  cords  of  wood,  how  many  solid  feet  ? 

10.  Reduce  3058560  cubic  inches  to  tons  of  round  timber. 

11.  Reduce  1  tun  to  gills. 

12.  Reduce  47  barrels,  16  gallons,  4  quarts,  to  pints. 

13.  In  17408  pints,  how  many  bushels  1 

14.  In  5927040  minutes,  how  many  weeks? 

15.  In  27894  seconds,  how  many  degrees  ? 

16.  In  7  lb.  11  oz.  3  pwt.  9  gr.  of  silver,  how  many  grains? 

17.  Reduce  45681  grains  to  pounds. 

18.  In  12  tons,  15  cwt.  1  qr.  18  lb.  4  oz.  12  dr.  how  many  drs.? 

19.  In  34  Ib^  0  oz.  6  pwt.  16  grs.,  troy,  how  many  lbs.,  avoir.  ? 

20.  In  9173841  nails,  how  many  yards? 

21.  How  many  barleycorns  in  360  degrees  ? 

22.  In  4755801600  barleycorns,  how  many  degrees  ? 

23.  In  30539520  inches,  how  many  miles  ? 

24.  In  96  square  miles,  how  many  square  feet? 

25.  In  1784217600  square  feet,  how  many  square  miles  ? 

26.  In  96  A.  2  R.  16  po.,  how  many  square  feet  and  inches  ? 

27.  In  622080  cubic  inches,  how  many  tons  of  round  timber  ? 

28.  In  1^  tuns  of  v;ine,  how  many  gills?  and  cost  at  6Jc. 
per  gill  ? 

29.  Reduce  12528  pints  to  hogsheads. 

30.  In  87^  bushels  of  wheat  how  many  pints  ?  and  value  at 
3.5c.  per  pint? 

31.  Reduce  475047465  seconds  to  years. 

32.  In  1020300'',  how  many  degrees  ? 

33.  Reduce  9s.  13°,  25',  to  seconds. 

34.  How  many  steps  of  30  inches  are  contained  in  95  miles  ? 
and  how  long  would  it  require  a  man  to  walk  it,  at  the  rv^  tf 
8000  steps  per  hour  ? 

35.  Reduce  \  and  yyg^  to  decimals. 

36.  Reduce  |H||-H-|f  ^^  a  decimal. 

37.  Reduce  .21  pints  to  the  decimal  of  a  peck. 

38.  Reduce  .7  drams  to  the  decimal  of  a  pound. 


264  PROMISCUOUS  QUESTIONS. 

39.  Reduce  73  feet,  96  inches,  to  yards. 

40.  Reduce  17  h.  6  m.  43  sec.  to  the  decimal  of  a  day. 

41.  Reduce  3  cwt.  0  qr.  12  lb.  7  oz.  12  drs.  to  the  decimal 
of  a  ton. 

42.  If  1  yd.  cost  D2.5,  what  will  .8  of  a  yard  cost  ? 

43.  How  many  acres  in  4  fields  ;  the  1st  contains  12  acres, 
2  roods,  38  poles  ;  2d,  4  acres,  1  rood,  26  poles ;  3d,  85  acres, 
0  rood,  19  poles  ;  4th,  57  acres,  1  rood,  2  poles? 

44.  In  2  casks  of  wine,  one  of  89  gallons,  17  quarts,  the 
other  120  gallons,  9  quarts,  how  many  barrels  of  31-^  gallons, 
and  how  many  gallons  remain  ? 

45.  How  many  yards  in  the  four  following  pieces  of  cloth  : 
1st,  49  yards,  2  nails  ;  2d,  148  yds.  3  qr.  1  na.  ;  3d,  9  yds.  7 
qr.  8  na. ;  4th,  6  yds.  1  qr.  0  na.  ? 

46.  How  much  paper,  f  of  a  yard  in  width,  will  be  required 
to  paper  the  4  walls  of  a  room,  whereof  2  walls  are  36  feet 
long  and  8  feet  high,  and  2  are  28  feet  long  and  8  feet  in 
height,  deducting  15  square  yards  for  windows  ? 

47.  Multiply  9  cwt.  3  qrs.  27  lb.  12  oz.  by  7. 

48.  How  much  water  will  be  contained  in  96  hogsheads, 
each  containing  62  gallons,  1  qt.  1  pt.  1  gill  ? 

49.  If  7  A.  20  po.  cost  D94.78,  what  will  4  A.  27  po.  cost  ? 

50.  If  19  hhd.  4  galls,  molasses  cost  D147.5,  what  will  37 
hhds.  19  qts.  cost  ? 

51.  What  is  the  value  of  (60-T-2-4x9  +  7)x(12+8-r4- 
3X2)? 

52.  What  is  the  value  of  (7-4)  X (12X3-2X7)^-3X11? 

53.  What  is  the  value  of  ^  j-f%f  g"  °^  Ts  • 

54.  What  is  the  value  of  (f+i  of  |.)^(A-2)  ? 

55.  Multiply  f ,  31  5,  and  ^  of  f ,  together. 

56.  Multiply  f,  f,  6f ,  5|,  and  12,  together. 

57.  What  is  the  value  of  f  of  f-rf  of  |  ? 

58.  What  is  the  value  of  4f -^1  of  4  ? 

59.  In  13f  of  a  cwt.  how  many  ^^  of  a  cwt.  ? 

60.  Divide  147_t.  cubic  yards,  by  13^ K- 

61.  2/  90764876.920097. 

62.  Y  690897624jff8- 

63.  2V768480^|H-,. 

64.  2y  8409709485.70904. 

65.  3/  .0496742189. 

66.  %/  97890985708|76, 

67.  3j/  478096847.09607. 

THE    END. 


\ 


lOi 


RECOMMENDATIONS. 

The  following  recomm<?ndation  is  from  Thomas  H.  Burrowes,  ] 
late  able  and  talented  Superintendent  of  the  Common  Schools  of  t 
— and  who  may  very  properly  be  termed'  !?«e  ** father  of  our 
School  System;'^ — 

*'I  feel  pleasure  in  expressing  the  opinion  that  the  Cotiunbian  Calculator, 
by  Mr.  Almou  Ticknor,  is  a  most  valuable  school-book.  The  adherence  to 
our  own  beautiful  and  simple  decimal  system  of  money,  and  the  exclusion 
of  the  British  currency  of  pounds,  shillings  and  pence,  which  Morris  one  of 
its  cliief  differences  iVom  other  Arithmetics,  I  consider  a  decided  and  valu- 
able imi:rovement.  It  always  appeared  to  me:  useless,  if  not  worse,  to  puz- 
zle the  beginner  in  A*rithmetic  with  questions  in  aiiy  other  money  than  our 
own,  at  a  time,  too,  when  the  unavoidable  jjitricacies  of  the  Science  are 
sufficing* ly  numerous  and  difficult  to  task  all  his  patience,  and  wKen  thfe 
teacher's  chief  object  should  be  to  excite  and  sustain  his  intere*^*  in  the 
study.  After  he  has  become  well  versed  in  the  principles  oi  Ai'-hmetic, 
imd  complete  master  of  all  calculations  in  our  own  coin;  it  is  nc>  rdy  pro- 
per to  give  him  a  knowledge  of  those  of  oihc:  lands,  but  it  v1*il  be  found 
practicable  lo  do  so  in  one-tenth  of  the  time  requisite  for  that  v  ;p'i:  e  at  an 
earlier  period.  In. many  other  respects,  also,  the  Coluiijfcian  «J.4k5tjlator  is  a 
superior  work,  and  I  therefore  cordially  recommend  it. 

Ti.^MAsH.  E-:..     r.Es." 

Lancaster,  Nov.  26,  1847. 


3f| 


(From  the  Carbon  County  Gazette.) 
Messes.  Editors  :  Among  the  latest  systems  of  Arithmetic  which 
been  presented  to  the  American  public,  and  which  are  destined  JVcni 
rior  excellence  to  supplant  earlier  school-books  in  this  department  oi 
cation,  it  has  seemed  to  me,  that  a  notice  of  the  "Columbian  Golciilr 
in  your  columns  would  subserve  the  interests  of  our  county  cf- 
schdols.  As  its  name  indicates,  it  discards  the  sterling  calculations 
in  use  in  our  colonial  relations  with  Great  Britain,  and  which  constitu* 
body  of  the  Arithmetics  in  use,  adopting  in  their  stead  computation* 
exercises  wholly  in  American  money,  and  nearly  exclusive  of  ft 
weights  and  measures,  except  for  the  convenience  of  reference. 

To  this  peculiar  feature,  so  desirable,  so  proper,  and  so  patriotic! 
work  sacrifices  nothing  essential,  but  in  perusing  it  Mr.  Ticknor'B  I^t 
must  be  regarded  as  furnishing  an  era,  ironi  which,  if  real  utility  be 
aim,  future  writers  or  compilers  will  date  one  of  the  most  signal  reforii} 
the  method  of  calculation,  and  follow  in  this  truly  national  v/akc.     For  ^ 
use  of  our  common  schools,  this  Work  commends  itself  before  most  O' 
rivals,  in  the  multiplicity  of  its  examples,  always  in  i^ro}:  ortion  to  th^*^- 
tical  nature  of  the  rules ;  in  the  immense  variety  presented  to  the  sM 
and  in  the  adapting  of  explanatory  exercises,  and  cross-t  Aaminatid# 
fixed  and  annexed  to  the  several  rules;  in  the  early  introductioi:'^^ 
tions,  as  important  and  necessary  as  units  and  integer-,  aud,  in  a  ic^ 
accurate  inductive  system,  should  precede  them  as  their  component 
in  the  good  arrangement  of  the  subject,  and  the  strict  test*,  adopted 
vent  superfifciality ;  and  last,  but  not  least,  a  brief  but  iidmiryble  :ih^ 
mensuration,  with  a  view  to  aid  the  mechanic  in  hisealcuiav'ona:  for  \> 
reasons  we  hope  soon  to  see  it  upon  the  desks  of  our  common  schools, 
the  place  of  inferior  books.  I.  H.  Siewers, 

Frincipal  of  the, High  School^  Carbon  County ^  Fa. 

Another  eminent  teacher,  of  Connecticut,  writes:  "1  should  judge  tW  ^_ 
by  its  systematic  arrangement,  perspicuity  of  explanation,  and  above  aU,\i^  jj*:,  I 
practical  adaptation  to  the  currency  of  this  country,  its  claims  are  euperjor    fc; 
^0  any  other  work  of  the  kind  extant,  and  eminently  worthy  of  its  nitmt.   f 
and  author.^*  '  ^ 


